As we all know, the collapse of the Byzantien Empire was a historical fluke and frankly the Caesars should be colonizing Epilson Eradani right about now.
But after Manzikert, the Empire's situation looked pretty damn grim. By 1077 the Turks had established themselves in Nicaea; Robert Guiscard was marching east; the Pechenegs crossed the border; etc.
So, let's suppose that in 1082, Alexius Commenius dies mysteriously. It's not poison, but in the ensuing knife fight the empire is paralyzed. Robert Guiscard can march on Constantinople; but I really don't think he can keep it even if it falls.
Wither Rome?