Then, suddenly, all three go berzerk with several huge stacks and completely slaughter their intended enemies.
I've never done an analysis of why this happens but I believe that at a certain point, any surviving Roman faction controlled by the AI gets a huge money boost. If one can believe the charts, I've seen all of them go from having 10k denarii or less to well over 100k. And this happens even if I have every single Roman port blockaded, which I've done on occasion. Not sure how to prove it, though....