It is 1025, and Basil II the Bulgar-Slayer has died. Unlike in OTL, he has capable successors--several "Good Emperors"--in command of his 250,000-man army over the next few decades. During the reigns of these powerful emperors, the Eastern Roman Empire destroys most of its rivals, playing them off against one another as the Roman Emperors were so keen to do. By the 1200s, the entire Mediterranean basin is ruled from Constantinople, and Mare Nostrum is restored. But this time, Greeks are in command.
In a scenario like this, how do the Byzantines administrate their western territories? Would they be granted to exarchs? Strategoi? Dukes? Feudal lords?
Would the feudal structures of the west be maintained for a few decades, while divide and conquer strategies are used to destroy the power of local lords and build up imperial power? Or would the nature of governance in areas like Italy, Spain, and France become more imperial than feudal as soon as Byzantine rule extends to those areas?
Would the resurgent Byzantines be able to prevent the Great Schism and reconcile the differences between the Western and Eastern churches? Perhaps if the Holy Roman Emperor tried to gain undue power in Italy, against the wishes of the Pope, so the Pope actually tries to appease the Eastern Emperor? (In fact, calling the First Crusade was partially inspired by a desire to improve relations with Byzantium, to act as a balance against the reckless Normans).
Is it more likely that the expansion of the Byzantines would lead to western rulers trying to gain the favor of the powerful eastern Emperor, either to oppose the Holy Roman Emperor or to retain their privileges against a steadily expanding Roman Empire? Or would the Catholics band together against Orthodoxy, leading to a more violent schism and a religious war?
How would Byzantines treat Romance languages and Church Latin?