HoMM3 HD is a mod (add-on) for the game Heroes of Might and Magic 3.

There is a suspicion that the problem arose (or worsened?) with the latest HD update (to version 3.47) – already, 2 local multiplayer games with a stable network connection ended with “desynchronization” issues during intense battles (restarting doesn’t help – the glitch appears in the same place).