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Since everyone loves Byzantine timelines around here:
So, we all know that the Latin Empire was a dismal failure, lead to the fragmenting of the Byzantine State, and lead to the eventual rise of the Ottomans. Fair enough.
But, if there anyway for the Latin Empire to consolidate its hold on the former Byzantine lands and creating a lasting state? Also, what would the effects of this state be, both in Eastern Europe and the West? Finally, how would the religious disputes play out, and what would the Latin state have to do in order to win over the Orthodox population (short of converting, of course; that seems unlikely).