AHC/WI: Greek Influenced Hungary

I'm not exactly sure what this is, so I'm going to call it both; can you get the Kingdom of Hungary to be more influenced by the Byzantines (rather than say, the HRE as in OTL), and, if so, what would the consequences of this be?

It can have any POD, but I assume it would be prior to Kings Geza and Stephen, who converted Hungary in OTL, but also after the Magyar migrations, as I want Hungary, not a Avar or Roman Pannonia.
 
Not sure about an orthodox convertion, but what about a Byzanto-Hungary under Bela-Alexios, with POD being Alexios II Komnenos either not born or being born as a girl?
I am not sure if such a union was possible or if it could last after OTL Bela III, but if it could it surely would result in a much more closer ties between Hungary and Byzantium.
 
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