Could the Turks and Greeks unite and reform the ERE after the 4th crusade? By that I mean a state in which the majority of people are either Turkish, Greek or a mix of both (all these groups living mostly peacefully with each other) conquers Constantinople and revives the Roman Empire. The religion of such a country doesn't matter, you just have to make it work.
The best scenario I could think of that happens before the Ottomans (I don't know much about them) is that the Latin Empire wins a pyrrhic victory at Poimanenon but due to a defeat against Epirus aren't able to further advance, and then the Mongols also experience a pyrrhic victory at Kose Dag, allowing them to conquer Eastern Anatolia, but not much else. The two groups would then be forced to cooperate, after a bloody war that is.
The best scenario I could think of that happens before the Ottomans (I don't know much about them) is that the Latin Empire wins a pyrrhic victory at Poimanenon but due to a defeat against Epirus aren't able to further advance, and then the Mongols also experience a pyrrhic victory at Kose Dag, allowing them to conquer Eastern Anatolia, but not much else. The two groups would then be forced to cooperate, after a bloody war that is.