In 1204 the 4th Crusade conquered Constantinople and set up the Empire of Romania (Roman land) which is usually called the Latin Empire of Constantinople, and the crusaders grabbed as much land as they could for various fiefs, while various Roman lords seized control of various provinces and some of them proclaimed themselves emperors of the Romans.
So in about 1229 there were:
1) Empire with capital at Trebizond,
2) Empire with capital at Nicaea,
3) Empire with capital at Thessalonika,
4) Latin Empire,
5) Holy Roman Empire,
6) Bulgaria, whose monarch used the title of "Emperor of the Bulgarians and the Romans",
7) the Turkish Sultanate of Rum (Rome).
8) Wales, where the rulers of Gwynedd may have claimed to be Roman Emperors as successors of Constantine III.
Most people will not consider all of them to be Roman Empires.
From 1355 to 1371 there were:
1) Empire with capital at Trebizond,
2) "Restored Byzantine Empire", the former Empire with capital at Nicaea,
3) Serbia, whose ruler used the title of "Emperor of the Serbians and the Romans".
4) Thessaly, whose ruler used the title of "Emperor of the Serbians and the Romans".
5) Bulgaria-Tarnovo, whose ruler used the title of "Emperor of the Bulgarians and the Romans".
6) Bulgaria-Vidin, whose ruler used the title of "Emperor of the Bulgarians and the Romans".
7) Latin Empire.
8) Holy Roman Empire.
Most people will not consider all of them to be Roman Empires.