This sounds like the old RTW multiplayer bug.
I often, if not always, had this problem back when Rome was just released and I played it alot in LAN, but we always blamed our connections or occasionally speculated that maybe it was some kind of odd trick of the engine to reduce or make lag less visible gone awry.
Hadn´t tried a LAN battle for years, and never in EB, so I decided to do one with a friend some weeks ago and the same happened again, although not before late in the battle - we didn´t notice it before at least - and the difference between what was shown on our two screens was limited yet clear.
In addition to a notable delay in the battle-progress on the PC not hosting, the AI seemingly made up for this by taking control of the host´s forces on the non-host´s end and conversely until at some point somehow the two were more or less syncronized again but with dissimilar experiences of the battle.

But I remember vividly the frustration of differing battle outcomes and when the differences became apparent early on the notion that you were really playing two seperate battles.