What's the latest realistic PoD for a surviving Byzantine Empire that contains the Pentarchy (Alexandria, Antioch, Jerusalem, Rome (south and central Italy)? Would the 12th century (before the sack) be doable? Or perhaps all the way back to Manzikert? I'm assuming that the Byzantines manage to integrate a substantial number of Muslims (as well as catholics) into the empire in the same way that the Ottomans managed with Christians.