As far as I know, there are no countries calling themselves Communist that have stayed real democracies, even though Marx wrote that he wanted democracy in Communist society. However, there are quite a few socialist countries, and some are effective democracies. Are there any socialist, real democracies that were established by Marxists?
There aren't a lot of extant nations that consider themselves explicitly Communist, Marxist, or Marxist-Leninist. Since the question asked about countries founded by Marxists, I assume it's not asking about some country where a Marxist party happens to have won some elections; that said, I also assume the question isn't asking about a country that was founded by Marxists but transitioned to some other form of government.
In any case, only a handful of states consider themselves Marxist. Per Wikipedia:
At the turn of the 21st century, China, Cuba, Laos, North Korea and Vietnam remained the only officially Marxist–Leninist states remaining, although a Maoist government led by Prachanda was elected into power in Nepal in 2008 following a long guerrilla struggle.
Cuba, Laos, China, and Vietnam are all de facto one-party states. Laos and Vietnam seem to be de jure one-party states in the sense that I can't find evidence of even subordinate political parties being allowed. Their leaders sometimes state that they are democratic, but that's dubious considering the lack of competition. North Korea is a hereditary dictatorship. Cuba does not have the formal bans on non-majority parties that Laos, and Vietnam do and the members of the legislature (the Asamblea Nacional del Poder Popular) are elected. However, its claim to democracy is doubtful: non-Communist parties are subject to persecution and the primacy of the Communist Party is explicitly described in the constitution and its first leader held office for almost 50 years with considerable executive power and was succeeded by his brother. Further, the candidates for the Asamblea Nacional are essentially picked, in large part, by organizations affiliated with the Communist Party.
China is a de facto one-party state, where the Communist Party is given supremacy by its constitution. However, it doesn't have formal bans on all "opposition" parties, and a member of the China Zhi Gong Party was appointed to a ministerial post in the last decade or so. Members of non-CPC parties hold a surprising number of seats in the legislature. And relative to Cuba, it's had more turnover in leadership. It's not particularly democratic: needless to say, as a matter of fact and law the CPC is the only real influence, and Xi has been trying to accrue more power to himself personally, but given the previous may be the most democratic of the five explicitly Marxist states mentioned here.
It's worth noting that many Marxist and non-Marxist commentators would dispute the claims of the some of these states to Marxism: for instance, the state-run companies in China can accumulate profits rather than redistributing them to the populace. Similarly, some of the parties that don't claim to consider themselves Marxist might be viewed as more Marxist by some Marxists.
If we extend our view to countries that didn't start out as explicitly Marxist, but which now have Marxist leaders due to elections, there aren't that many. Obvious suspects like the Movement for Socialism–Political Instrument for the Sovereignty of the Peoples, whose leader Evo Morales currently presides over Bolivia, or the African National Congress in South Africa, don't seem to identify as Marxist. The ANC considers itself social-democratic, although it has praised the Chinese Communist Party, and despite the name, MAS-IPSP doesn't seem to consider themselves even very socialist, let alone Marxist or Communist—at least not enough to mention it in the history portion of their website, despite the fact that Marxist politicians seem to have played a role in its rise. Commentators outside these and other parties, and even some members, may consider them Marxist or Communist, but they don't seem to view or present themselves this way, per the terms of the question.
That said, I would submit that Nepal qualifies according to this second definition. The president, Bidhya Devi Bhandari, was a member of the Communist Party of Nepal (Unified Marxist–Leninist), which has both Communist and Marxist right in its name, up to the point of her election. I assume she was obligated to discontinue her position in the party upon taking office, but she doesn't seem to have greatly altered her opinions. The Prime Minister, KP Sharma Oli, is one of the current chairs of the Communist Party. Although there were various complicated mergers and dissolutions, it appears they also still hold a majority of legislative power. Nepal's most recent constitution is still new, but it both expresses a commitment to socialism (although, seemingly, not to Marxism or Communism), and to democracy:
Expressing commitment to create the bases of socialism by adopting democratic norms and values, including peoples' competitive multi-party democratic governance system, civil liberty, fundamental rights, human rights, adult franchise, periodic elections, complete press freedom and an independent, impartial and competent judiciary, and the concept of rule of law.
Still, this is a country that established socialist democracy not specifically based on Marxism, in which a Communist Marxist-Leninist party happens to hold the political power, at least at the moment, so it doesn't strictly meet the terms of the question.
Yes, it is called XXI century socialism. Venezuela is one of those countries. Here is a summary from Hugo Chavez himself.
We will continue calling it Bolivarianism, but it is Socialism. The goal is Socialism, it is the only alternative to Capitalism, I only believe that we must reinvent Socialism, reinvent a Socialism adequate to our reality and our XXI century.
Here is also a video where he make this claim.
He also states that "I read Marx" and that "Marx was right".
When Chavez arrived to power Venezuela was a democracy and he was elected by popular vote. The problem is that the institutions were slowly destroyed one at a time. Socialism is incompatible with democracy and the rule of law. The more you have of one the less you have of the other.
You have to understand that a lot of the socialism you see today is directly based on Marx and Engels. So it depends on what you understand as Marxism, Communism and Socialism and where you draw the line. That line is not always clear. Lots of authors that followed Marx, based their work on him but they criticized materialism as Marx conceived it (in part because it is flawed and too rigid). The Frankfurt School is one of those groups. They are what we know today as Cultural Marxism or Neo-marxism.
The Frankfurt School perspective of critical investigation (open-ended and self-critical) is based upon Freudian, Marxist and Hegelian premises of idealist philosophy. To fill the omissions of 19th-century classical Marxism, which could not address 20th-century social problems, they applied the methods of antipositivist sociology, of psychoanalysis, and of existentialism.
As stated earlier democracy and the rule of law are incompatible with any form of Marxism, both materialistic and cultural. In theory the state should not exist in communism therefore all that humanity knew in the last century was socialism not communism even when those countries "called themselves" like that. Real Marxism in any of its forms relies on the power of the state. The more the state grows the less democracy you have because the state owns the monopoly of violence and use force as the means to enforce its view into the citizens. The rule of law uses the state as a simple regulator to allow freedom of choice and the state itself is not free from the penalty of the law.
The rule of law is defined as: "The authority and influence of law in society, especially when viewed as a constraint on individual and institutional behavior; (hence) the principle whereby all members of a society (including those in government) are considered equally subject to publicly disclosed legal codes and processes.
This section written by John Locke is relevant and it is what we understand today as rule of law
Freedom in society means being subject only to laws made by a legislature that apply to everyone, with a person being otherwise free from both governmental and private restrictions upon liberty
In socialism the state changes the law to enforce the particular view of some groups into the society therefore you are not free from government control and the rule of law does not exists.
The turning point in this matter was a significant event that occurred in 1989. The fall of the Berlin wall and the collapse of the Soviet Union proved to the world that traditional Marxism and Socialism in general was actually worse than Capitalism. The fact that the soldiers in the wall were there to prevent citizens from leaving instead of just guarding the borders and also the gulags present in all Soviet Union were proof enough but this was just the end of traditional Marxism as we know.
Following the events of the Soviet Union, socialists intellectuals all over the world started to review their theories to seek alternatives to impose their view of socialism. The so called São Paulo Forum is one of the most relevant examples. They decided that the way to destroy the Capitalism was to use democracy to reach the government and slowly dismantle the rule of law. This is the part where Venezuela enters the scene. The transition to full socialism started by Chavez was not completed because of his death. The Maduro's regime revealed to the world the complete disaster Venezuela, one of the richest countries of Latin America, was forced to endure. Even when is not a socialism like the Soviet Union you can clearly see for yourself the ruined state of the rule of law in this country and the advance of Socialism everywhere. It's unlikely that after the fall of the Berlin Wall any socialism can destroy Capitalism in any country today in one blow. What you have is a slow transition and a "more" or "less" socialist country.
As for Marxism in Venezuela, Hugo Chavez started nationalizing all properties and would eventually ended up in a system were the State owns every means of production just like traditional Marxism. Maduro continued this tradition. Again Chavez death was a huge blow to this course of action. The problem is that today you cannot openly claim that you follow Marx if you don't have a hold on power because after the Gulags the people reject the doctrine implicitly. Heinz Dieterich was supposedly one of the main authors of this current.
This form of socialism is revolutionary in that the existing society is altered to be qualitatively different, but the process itself should be gradual and non-violent, instead utilising participatory democracy to secure power, education, scientific knowledge about society and international cooperation.
In real life Chavez was more influenced by Castro than Dieterich which lead to the later breaking up and claiming that Venezuela was not socialist, a statement that is constantly disputed by the people on power. Here is a quote from Castro
Marxism taught me what society was. I was like a blindfolded man in a forest, who doesn’t know where north or south is. If you don’t eventually come to truly understand the history of the class struggle, or at least have a clear idea that society is divided between the rich and the poor, and that some people subjugate and exploit other people, you’re lost in a forest, not knowing anything.
Most of the sources referring specifically to this new form of socialism are in Spanish. The book The Populist Deception: why our countries are ruined and how to rescue them is one of the best references. I hope it's available in English soon. There are other forms of cultural socialism but they are too broad and more difficult to verify than XXI century socialism which is still a little bit materialist and more similar to classical Marxism.
Those advances were made possible using the precepts of Gramsci (also socialist) to use cultural penetration as the means to destabilize and destroy Capitalism values and cultural advances. This is why you have today Bernie Sanders in the US and most people think that Neo-Liberalism is bad but they cannot describe you what Neo-Liberalism really is. Nobody agrees at which point a country started to be openly socialist and stopped being Capitalist because Socialism had a tremendous influence in Western Culture despite the Soviet Union disappearing.
Marx believed that after a direct democracy (aka dictatorship of the proletariat) period, socialism could eventually happen, followed by communism. Since (as far as I know), not a single dictatorship of the proletariat has turned out well, it is very hard to say that a country implement socialism in the Marx way.
Chile, while Salvador Allende was the democratically elected socialist president, was becoming a socialist country. But the CIA and Henry Kissinger had the very undemocratic idea of stopping this experiment. Augusto Pinochet with American help overthrew Allende. The country returned to capitalism and stopped being democratic for many years.