
Byzantine Empire in 1101 after the First Crusade
I modified this map, inspired by NeimWiki

I modified this map, inspired by NeimWiki
I was curious about Sicily, specifically from 668 to 827 AD. Was this the best place to live in the Empire during this period?
So Constans II is murdered in Syracuse in 668. As far as I can tell, no Roman emperor would ever again set foot on Sicilian soil. After that, it appears that Sicily routinely ignored military orders from Constantinople. This is probably why you never heard about some heroic relief army coming to save the empire during its many crisis's. Instead, the Theme of Sicily appears to have spent its resources reinforcing local defenses and dealing with pirates and raiders (which Sicilians had been dealing with for centuries).
In religious terms, the Sicilian people and church sided with the Pope over Iconoclasm and the evidence is that the local Roman strategos simply did not enforce the constantly changing religious edicts from Constantinople.
Tax money also didn't flow back to Rome. Instead, Vatican records showed that various popes negotiated for grain and tribute directly with the strategos in Syracuse rather than with Constantinople. Sicily used its resources to conduct similar, independent foreign agreements with the Arabs, Franks, Lombards, and others. This (along with the anti-pirating) took advantage of Sicily's strategic location as a trade route.
When all else fails, with the rest of the empire seemingly in flames, Sicilians were left with a peaceful, prosperous, self-sufficient Mediterranean island with great weather.
So if you looked at a map from this period, it would show that Sicily was part of the Roman Empire. But for all practical purposes, did they just have a de-facto independent state with a nice climate?
Of course, the party ended when Euphemius got too big for his britches in 827. The local Sicilians turned on him because he was messing with a seriously good situation. Sadly, he fled to Africa and sparked an Arab invasion of the island.