I've seen a lot of timelines and what-ifs describing a Byzantine resurgence at various points in their history. But what if the Byzantine Empire collapsed even earlier than in the OTL? Is it possible that a stronger and more stable alt-"Latin Empire" would have taken its place?
Unnecessary background: thanks to the disastrous leadership of Guy of Lusignan at the Horns of Hattin, the Kingdom of Jerusalem fell to Saladin in 1187. Possibly necessary background: the Byzantine Empire could care less about Jerusalem or Saladin, as their major focus was on Asia Minor and their primary rival was the Sultanate of Rum. Shortly after the fall of Jerusalem, the Byzantine Empire concluded a treaty with Saladin's nascent Abbuyid sultanate against the Turks.
Needless to say, this didn't go over well. Frederick Barbarossa sought and received permission to bring his armies across the Byzantine Empire by land, but once they arrived the reigning Emperor Issac II Angelos apparently changed his mind. Barbarossa ended up seizing the city of Philippopolis, and defeated a Byzantine army of 3,000 sent to relieve it. Eventually Issac II reconsidered and allowed Barbossa to pass through.
In the OTL, Barbarossa was apparently willing to let bygones be bygones, and led his Crusading forces against the Turks in Asia Minor. The Sultanate of Rum suffered several defeats, most notably at the Battle of Iconium where Barbarossa sacked and seized the Turkish capital city at Konya. The German Crusaders continued through Asia Minor, but Barbarossa died while crossing a river and the army dissipated before reaching its destination in the Holy Land.
But what if Frederick Barbarossa hadn't been so conciliatory with Issac II? The Byzantine Empire already had a strained relationship with western Europe, and had been recently weakened by war with Norman Sicily (which had sacked the city of Thessalonica). Issac II had secured his throne four years earlier with a victory at the Battle of Demetritzes, which drove the Normans off, but the Byzantine Empire was almost continually on the brink of disaster.
So what if Barbarossa hadn't agreed to help the Byzantines? What if he reconsidered and found that his position was a remarkably solid one -- with a larger army with many allies (notably including a Hungarian contingent) perched outside the capital city of a much-weakened empire? Barbarossa himself was the Holy Roman Emperor, and was famous for his political and administrative acumen as much as his military prowess. What could he have done, with the resources of the Byzantine Empire under his command?
I don't have the requisite knowledge to make a timeline of this on my own, but I consider it a truly fascinating divergence point in history that I'd love to see others continue. What could he have accomplished, and what would history look like had Frederick Barbarossa chosen differently?