Inevitably, this resulted in Bohemond proclaiming himself as Prince of Antioch in the aftermath of the First Crusade.
It wasn't
that inevitable. If Stephen of Blois hadn't chickened out and told Alexios that the Franks were screwed, Alexios might have kept marching east, and I think it's very unlikely that Bohemond would have kept the city if Alexios was on the scene. The other Frankish leaders were not exactly excited at the prospect of leaving Bohemond with the city.
However, what might have turned out differently if Alexios had agreed to Bohemond's initial request and appointed him as the Domestic of the East, i.e., the commander of the eastern armies ?
I'm not sure, but based on what we know of Bohemond I'm guessing the word "treachery" would be involved. I think you're more likely to end up with "Roussel de Bailleul 2.0" than "Bohemond, hero of the empire."
I suspect the
immediate consequence of this would be to arouse the jealousy of the other Frankish leaders, because Bohemond has now been given a high position by the emperor and will probably try to assert his authority as leader. These guys were never on great terms with one another and it's entirely possible this blows up the whole enterprise in some unforeseen way. Even if they manage to keep things together as far as Antioch, there will probably be continual friction between the Franks who believe they're making a righteous pilgrimage for the Pope and Domestic Bohemond who is now apparently doing this as a
job for the
emperor.
Post-Antioch (assuming all else remains the same), the Crusaders probably spend a lot less time fucking around - IOTL they basically went nowhere for six months because of bickering about what to do with Antioch and general rivalries among their leaders. If Bohemond is the Domestic then there's no real question that he gets Antioch (in the name of Alexios, of course), and the Crusaders presumably don't have anything to keep them from moving south. You might also avoid the siege of Arqa, another massive waste of time, although perhaps Raymond is still dead set on getting Tripoli. It's worth noting that the Fatimids only captured Jerusalem from the Turks
after the Battle of Antioch, I believe in September of 1098, so if the crusaders are
really fast they might stumble across the city right as the Fatimids are besieging it. If al-Afdal and the Franks meet below the walls of the city, both with the same objective in mind... well, I'm not exactly sure how that plays out. "Badly," I assume.
