Crazy enough, this question appears to not have been posted for its own thread before.
What if the Sultanate of Rum and the Latin Empire, which had practically split up the Eastern Roman Empire, both survived and had become the new norm for the region, with both surviving until at least the end of the 1500s.
I believe that, by this time, the Latin Empire would be Hellenized but still Catholic (seeing as how that's what defined them more than anything) and the Sultanate of Rum would also be Hellenized but still very Turkish and Muslim like IOTL.
Also, I could see the Catholic and Islamic Worlds seeing their respective states as being the true Successor to Rome.