WI: A successful Crusade?

Shortly after the fall of the last Crusader city of Acre, Nestorian monk turned diplomat Rabban Bar Sauma travels under the command of Arghun Khan to Europe in search of a Ilkhan-European alliance against the Muslims. In OTL, it didn't work as the European nations were starting to enter into conflict again in too a bad way to pay heed to a new Crusade. But what if the Tenth Crusade had been called and this alliance was able to take over most of the Muslim world?
 
I haven't really thought on the ideas. After all, the idea recently came to my mind. But I guess that further cooperation with more tolerant Muslim dynasties like the Taifas or something and an attack from both the Ilkhanate on one side and the Crusaders on the other side may squeeze the Muslims out into the Maghreb and Arabia or something. But again, I've only thought on the basic outline of a possible scenario and don't yet have an idea.
 
Well, at this time, the Il-khanate isn't at its peak strength, and it is very unlikely that the Europeans will be co-ordinated enough to take any core Muslim regions such as Egypt or Damascus. Just maybe they might take Tunis or somewhere, but there is not much of a chance that they hold it in the long run.
 
I suppose the question has to be asked: How are the Europeans in better shape?

There are ways they could be in better shape - enough so to launch a great crusade - but they'd need a point of departure some time back.
 
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