Do we know what "hardness" of Brexit people actually wanted in the referendum, if there had been other choices available?

Question

As worded, the referendum was between two choices - leave or stay. But realistically, if there had been a mechanism for it, there would have been other choices on the list, reflecting the nuances of future negotiations.

Perhaps a full menu of "Leave" options might have split the leave vote into these kinds of subdivisions:

• Leave with or without a deal
• Leave primarily because of promised benefits taken as realistic (the famous "£350m a week to NHS", or "X trade deals within a year" come to mind as enticements that may have compellingly swayed some voters)
• Leave with a deal only
• Leave with no hard border (customs union etc) only.

At present its not at all clear what "52% voted leave" actually represents, and how many of them wanted what "hardness" of leaving.

Do we have any data about sentiment and nuanced subdivisions, which might shed light on this, as at around the time of the vote?

Answer 1

Since you want this answered as close to the referendum date as possible, I did find one survey by What UK Thinks from the autumn of 2016 (the Referendum was held in July). It is pretty elaborate, so I suggest you read it all for the details, but its summary findings were:

Nine in ten people would like free trade with EU countries to continue.

But at the same time, as many as seven in ten (70%) think the UK should be able to limit the number of people from the EU who come here to live and work, including 55% of people who voted to remain in the EU.

The public is split [49% vs 51% overall] over whether Britain should accept freedom of movement of people in exchange for free trade with EU countries.

This split is even clearer when we look at those who voted to remain in the EU and those who voted to leave. [30-70 in the Leave vote vs. 70-29 in the Remain.]

Frankly this is not much of a Brexit model; more like a "why do you [not] want Brexit" broling down to the tradeoff between immigration and free Single Market access.

But this survey does find a rather prevalent "have the cake and eat it" attitude among the British public at this time:

In short, the apparently widespread wish amongst the British public to remain part of the single market in goods and services but to opt out of the obligations the EU’s freedom of movement provisions is not confined to those who voted to Leave. It is also to be found amongst many who voted to Remain. Such a deal would, it seems, be backed by a majority of voters on both sides of the referendum. But, as we have already noted, many political leaders in the EU have indicated that this is one deal that will not be on offer to the UK when the negotiations take place.

There's also a RAND study from February 2017 (which also has a later, longitudinal follow up.) This RAND study is actually synthesizing a Norway-like model of Brexit based on the responses. The conclusions of the first RAND study were that:

The British public want a deal. The British public want a deal on Brexit and are willing to compromise to get one. Netting out the positives and negatives, we find that the current situation of EU membership is worth about £14 per household per week more than leaving the EU with no deal.

People are more concerned with managing demand for public services than simply restricting freedom of movement. This is particularly true of those who voted to leave the EU.

People highly value having access to EU markets for trade in goods and services, but also would like the UK to be able to make its own trade deals. The British public place the greatest value on having the ability to make trade deals and retaining access to the Single Market for trade of goods and services after Brexit, more so than restricting freedom of movement, increased sovereignty and reduced EU contribution.

People value the Single Market access and the ability to make trade deals. The British public value the UK being able to make its own laws, but not as much as they value Single Market access or the ability to make trade deals.

People with degrees hold stronger views. Education level was the most important explanatory variable in quantifying people’s preferences. Overall, those with university degrees preferred closer ties to the EU. They were more positive about the value of freedom of movement for holidays and working and strongly disliked options with severe restrictions on freedom of movements, such as requiring a visa to travel to other European countries for holidays (and requiring other Europeans to have a visa to travel to the UK). They were also less sensitive to the level of EU contributions and held differing views about the importance of UK sovereignty over its laws, preferring options where the UK is subject to EU laws in environment, employment and trade.

People prefer a final agreement which is close to an agreement that is similar to the Norway model. Given the importance of making trade deals and retaining access to the Single Market, we find that the public place a positive value on a relationship like Norway’s current relationship with the EU, allowing for free trade with other countries while remaining within the single market, and accepting freedom of movement and some loss of sovereignty.

Do note that things like the backstop etc. were not in the public discussion sphere back then, so they didn't get included in the questionnaire. A longer version of seemingly the same study was published in July 2017. The follow up that I already mentioned conducted about a year later. Found pretty much the same. Alas, the Irish border problem isn't addressed by a plain "Norway model" like the one outlined above.

For more recent surveys on the more concrete (draft) Withdrawal Agreement between former PM Theresa May and the EU, see Is there a survey of the public opinion whether no-deal is preferable to the November deal?

Answer 2

Short Answer: No

Since the 2016 referendum offered a binary Remain/Leave choice, we cannot know for certain what people meant when they voted leave. All we can know for certain is that 52% of those who voted in the 2016 referendum wanted to leave the European Union in some way.

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That said, longer answer: Maybe

Since we cannot draw conclusions from the referendum itself, all we can look at is polls. These should be taken with a pinch of salt, given polling failed to predict the referendum result. Most of the following polls were found through What UK Thinks, an organisation that catalogues opinion polls:

2 YouGov polls, one from just after the referendum and one about a week later, show primary support for a simple trade deal, but a very split electorate.

This ComRes poll from about 2 weeks after the referendum shows that most people would accept the continuation of freedom of movement in order to retain access to the single market.

This poll from Greenberg Quinlan Rosner Research conducted immediately after the referendum again shows that people care less about controlling immigration than they do about retaining access to the single market.

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The conclusion I draw from this information is that many leave voters, at least at the time, wanted a deal that preserved access to the single market and were willing to sacrifice freedom of movement to get it. This is consistent with a larger study taken in early 2017, which found that people highly valued access to the single market, and were less concerned with freedom of movement than access to public services. That said, I must repeat we cannot concretely say that this is what leave voters wanted when they voted, because the binary nature of the referendum denied us this information.