Byzantines conquer most (or all) of Spain?

[Venetian Doge]

I would think that there would be plenty of Greek cultural influence left over (similar to Arab influence OTL) on southern Spain. However, I think Justinian will have to give it full autonomy considering its distance. While the increased profits might help the Byzantines overall, I concur with earlier posters that there won't be that many butterflies (except possibly an even easier Arab conquest? Since the Visigoths have to take more territory).

 

[Madhukar_Shah]

Are there possibilities for a restoration of the Western Roman Empire on those circumstances?

Most likely going to be a massive headache for the Byzantines- any western emperor, or Western Roman governor with sufficient power is likely to get himself proclaimed emperor and try and install himself in the east- that's something the Visigoths are never going to do. Apparently some people think the fall of the west represented the results of a deliberate policy by the east to get rid of this pool of pretenders.

 

[I HAVE BECOME GOD]

Well that's just fine isn't it? Convert and you get to keep your lands. Don't convert and your lands are up for grabs to those who did. Let the Goths tear each other apart while spreading the good word.

Purposefully starting an invasive large scale long term conflict all the way on the other side of the med when you have much, much bigger problems living in your backyard seems like poor administration.