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Yes, that's correct. Version 3.1 was installed on both computers. We were running two maps downloaded from the internet (without AI, so there was no need for 5.5utility), patched with MMH55_Mapmixer, and copied to the other computer. All the "rules" were followed (specifically, those regarding 8 skills, saves, etc.). The game was launched using 5.5x64 on Windows 10x64.
Initially, everything worked fine. The difficulty was set to hero level (it's more interesting without resources), and we chose the races and random starts ourselves. The actions always differed. However, as soon as the second month arrives, around the second week, everything falls apart – desynchronization occurs after any battle, regardless of whether it's with map-bound monsters or "weekly" ones. The result is always the same. Only if we don't fight and just move around, collect items, etc., does everything work normally.
Regarding EVE (2.7.1 and previously 2.6.2), there were no such problems at all. We used the latest version of EVE (in 2.6.2, I downloaded it separately because you can only use 8 skills in HotA; in 2.7.1, the standard version was used).
We haven't tried playing HotA with 5.5, and I don't see how there could be any desynchronization there.
I can send you the save files from both computers, as well as the profile from the documents folder. Maybe this will help in investigating the problem.
P.S. I forgot to mention that we always used mayors in the cities. Once we appointed them, they stayed there behind the walls