AHC: Keep Venice under Byzantine authority

For much of its early history, Venice was under Byzantine rule, both in name and in fact. Eventually, that authority became more nominal. Traditionally, their independence was recognized in AD 803.

So, the challenge is for them to be recognized as formally and effectively part of the Byzantine Empire for as long as possible. Ideally, avoid sending in troops to prop up a pro-Byzantine administration, though brief interludes of quasi-independence (say, the election of an anti-Byzantine Doge) are fine.

A more expansionist Frankish Empire, more successful on taking Byzantine or former byzantine territories in Italy, may reinforce the pro-byzantine faction in the city., critically if Carolingians manage to really threaten the city.

I would see the Duchy of Venice being still largely autonomous (as was the exarchate of Ravenna), but with regular byzantine influx, despite the fall of Carolingia.

At some point, however, its alligeance to Constantinople is going to have repercussion. If the city survives late Carolingians, and Byzantine navy decline in west face to Arabs, division (religious, but not only) between Latin and Greek churches would make a survival along Byzantines hard. I'm not sure that a byzantine Venice could last as such after the XIth century.

Either swallowed up by a neighbouring entity (as Amalfi face to Normans), maybe Aquilea Patriarcate; or either joining up with Latin church (meaning an independence de facto).