[Hungary AH] - Conversion to Orthodoxy?

See above. Instead of siding with the Pope, the Hungarians decide to look to Constantinople and enjoy an autocephalus patriarchy. Now Orthodox, how does this effect the Balkans and the areas around them? Do areas in what were once Yugoslavia more concretely become Orthodox? Would having a shared religion make the Hungarians and Byzantines more often allied? Might this spread Orthodoxy further into Central or Western Europe? Any other butterflies I might be missing?
 
An Orthodox Hungary would certainly help the denomination spread further in the Balkans, leading to a largely Orthodox Bosnia and parts of Croatia. There's also the possibility of spreading Orthodoxy to the Slavs living to the north.

A shared religion would improve relations on all levels, but it doesn't necessarily mean an alliance. I can see Hungary trying to expand to the south and wholly or partially take over the Byzantine legacy; a bit like Bulgaria or Serbia in OTL. An Orthodox "Empire of the Hungarians and Romans", from the Carpathian Mountains to the Aegean or Black Sea? (or both?) Of course, it depends on a lot of things. Hungary may fail to achieve this, or get distracted.
 
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