Having played the Byzantines for quite a while now with the 1.2 patch (Including for hour and hours on a laptop while crossing the Pacific in a plane) I will stand by my initial walk through on page one or two of this thread. Though I have modified it a bit.
The short version
Marry Anna to the Hungarians
Make a treaty with Venetians and wait for them to break it
Go after the rapacious Turks from turn 1
The object is to gain control of the south East corner of the map from Tunis to Zagreb. To do this you simply have to be wily enough to keep from being at war with more than two factions at once. Something that has changed with the patch is that it appears that there are now many more times when the AI will blockade a port or place a city under siege simply to gain favor with merchants or nobles. You can sometimes let them blockade your port or siege your city for two turns and find that they are more than happy forgive and forget. This is especially true if you can keep your reputation higher than despicable which is sometime a problem for take 'em and sack or exterminate 'em me
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