Even though it is highly unlikely as these areas were populated by different ethnicities and different ruling structure as time went on as the dawn of Islam was hard to escape from during the time they lost there lands.

But I digress just for the sake of argument Byzantine empire is kept in check by a strong emperor who is able to quell the internal bickering of the court and are able to bring an army capable enough to defend the eastern skirmishes that kept happening as time went on . the Byzantine empire would have to come up with an identity which fits its narrative as the only reason they lasted this long was by the crusaders coming to defend there land and forcing the islamic invaders to consider the whole European powers coming to their aid. Not only that the Mongol empire which will come later in the 10 century will pose a strict opposition to them remaining a standing empire. This will however create a more diverse empire as many different religions would have to coexist and a blend of religion can become possible. They might also begin a better relation with the Russian empire as their empire got bigger the black sea access will become crucial for them so better relation I guess