During another thread discussing the viability of a continued Mexican Empire I stumbled upon that fact that, while French troops were pulling out of Mexico, Napoleon III was actively pushing Emperor Maximilian to evacuate with them back to Europe. In our timeline, as we know, he chose to stay behind and fight a losing battle for his throne but it got me thinking; would it be possible for Maxy to instead negotiate with the Republicans to allow a peaceful exile for himself and his Conservative supporters on the other side of the Atlantic? Given the violent back and forth that had marred Mexican politics for decades, there's certainly some potential appeal to the idea of removing the ringleaders and military backbone of the ideological opposition to the Republican leadership, who could see this as an opportunity to finally settle the question of Mexico's government once and for all. And, if Santa Anna is any indication, such a move wouldn't be unprecedented.
If such an exile were allowed, where would Max end up settling? France is one possibility, but so would be taking up a position with his Habsburg relatives. Given his governing experience, family reliability\loyalty, and tested liberal-reformist views, there'd certainly be a place for him in the restructured Habsburg state that's emerging out of the crisis of the Austro-Prussian and Italian war loses and negotiations with the Hungarians and Bohemians