But what changes when your dealing with entities that exist in this weird geopolitical state, where the idea of it living by a nations rules often times gets called into question?
Companies that operate in a country must follow the law of that country.
What usually happens is that in each country you have the "local branch" as a separate company, that is owned by the parent (international) company. So, it is a local entity, legally equal to a "homegrown" company. This way, "E-int" legal obligations to that country are limited.
So, we have "E. Corp International"(E-Int) and "E. Corp Elbonia"(E-Elb).
What happens when you want to break up a toxic corporate trust when said trust is multinational in nature?
You just break it. The most basic would be dividing "E-Elb" into "F-Elb" and "G-Elb". "E-Int" still owns "F-Elb" and "G-Elb", so he has not lost its property. "F-Elb" and "G-Elb" would have a separate board and operate independently as completely separated companies. If they don't, they could be investigated and punished by collusion, price fixing... the same way that unrelated companies doing the same could be punished.
A step further would entail forcing "E-Int" to sell part of its shares on "F-Elb" or "G-Elb", or both. But that usually only is done in strategic sectors1 (and, if "F-Elb" or "G-Elb" are in one of those strategic sectors then "E-Elb" was in that sector too, so it should have had a mixed ownership before the breakup).
And the last step would be the takeover of all or part of the local company.
For all of those, "E-Int" or "E-Elb" should be able to fill legal procedures to the local law system to try to stop the procedure. Should that fail, "E-Int" (and maybe "E-Elb") could appeal to international organisms, by arguing that Elbonia is either unfairly targetting "E-Int" or that its laws are in violation of treaties that Elbonia is part of.
Even when the origin of the "toxicity" of the monopoly is outside the local country (for example, Microsoft monopoly case in the EU), the local laws may impose restrictions to try to fix it (Microsoft was forced to offer a way for users to easily change its default Internet browser).
In real life:
Things get more complicated, depending of the political and economical power of each side. E-Int could "convince" the governments of other countries with strong ties to E-Int to apply political or economic (trade wars) pressure2. It could even try to appeal to local justice system, but that remains highly controversial3, but of course the effectivity of hte local justice system will be highly dependent of the power of its country.
Military industries, facilities (power generation, etc.), mass media, etc.
Yet paradoxically, the European Commission, despite determining that the merger was anti-competitive in the commercial aircraft sector, decided to extract various concessions from the Boeing Company (hereinafter, Boeing) instead of blocking the merger. The Commission's retreat from Van Miert's earlier position was largely a reaction to President Clinton's admonition that an anti-Boeing/McDonnell Douglas merger decision could lead the United States to bring the matter to the World Trade Organization or to pursue "some options ourselves." The Boeing/McDonnell Douglas Merger: the European Commission's Costly Failure to Properly Enforce the Merger Regulation, page 2
3With the 2014 Argentinian debt restructuration, an USA judge tried to seize Argentinian assets in order to pay debt holders that did not accept the restructuration; while the ruling has been held by the USA legal system it has drawn a lot of international criticism, as it multiplies the legal risks of any operation.