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

I don't quite understand how you connect my question to the turn-based nature of strategy games. The lack of this option makes playing the game with more than two or three players pointless, as it takes too much time.
Heroes 5 is a prime example – AI players operate in parallel in real-time.
Moreover, even in HotA, in online mode, you can see your ally's turn in real-time.

Let me try to explain how I understand it.
When players take simultaneous turns and press the "end turn" button, they send data packets to each other containing their moves. That is, a simultaneous turn is a conditional concept; at the program level, there is no actual simultaneous turn – this is related to the game's original code.
In other words, we have a pseudo-simultaneous turn. And the interruption of the AI turn occurs precisely when inconsistencies appear in the data packets: this can be either an encounter or an intersection of hero paths, and other factors that are taken into account when "comparing data packets" (an example of such an inconsistency is that one player has already ended their turn and picked up a chest, but their opponent has not yet picked up this chest and may pick it up on that day, since the map update occurs at the moment of mutual turn transfer, as a result, the same chest is picked up twice – and this is an inconsistency, so the AI turn is interrupted, and the opponent has to redo their turn).
Can this be fixed? You can try it in your spare time – even in its current form, developing the AI turn is a complex and time-consuming process. It's not like sending data packets every millisecond... After all, this is not a real-time strategy game, but a game with a different original engine...
As for seeing your ally's turn, it's as if it's also our turn; that's why we're allies: we can't attack each other.