Sicilian Empire? Venetian Constantinople? Super Bulgaria? Check out "Basilicus Sicilia"
Here's another alternate-historical treat for you, my loyal readers. It's called "Basilicus Sicilia."
In our history, the Holy Roman Emperor Frederick II was one of the most fascinating characters. He was born and grew up in Sicily, the cosmopolitan heart of the Mediterranean. He spoke six languages and was a major patron of the arts and sciences. He became Holy Roman Emperor and ruled over Germany and much of Italy and was often at loggerheads with the Papacy, which excommunicated him repeatedly and at one point even claimed he was the Antichrist. He had a reputation as a religious skeptic (the truth of that is rather dubious given how the people writing about him were churchmen who had a political beef with him), but waged Crusades all the same.
In this timeline, Frederick's attempt to become Holy Roman Emperor is defeated by Otto IV, but he remains in power in Sicily and southern Italy. Still the genius he was in our history, he renounces his bid to be Emperor in exchange for Sicilian independence and large ransoms for the prisoners he's taken and instead begins a campaign of conquest in North Africa. A Catholic empire ruling over a large Muslim population is a tricky thing, but Frederick's upbringing in very diverse Sicily proves very helpful here. He takes direct control of parts of the Holy Land while Crusading and forces his nobles into submission, establishing himself as Europe's first absolute monarch. His successors continue his work, establishing a very interesting Mediterranean empire.
"Butterflies" from this event include:
*Bulgaria becomes a powerful Balkan state, causing the collapse of the Latin Empire of Constantinople. The Venetians end up with possession of the city itself, creating the position of "Junior Doge" to govern it.
*The Byzantine successor state of Trebizond reconciles with the Nestorian heresy common among the Christian Mongols and becomes a friend of the Ilkhanate. As a result, they are the first Christian state to use cannon.
*The Angevin Empire (England, much of Ireland, and much of France) remains intact rather than the infamous King John losing most of its continental possessions to the King of France. We still have something resembling the Magna Carta in this timeline, fortunately.
So if you're interested in another medieval Mediterranean might-have-been, check this out.