Ideas and suggestions.

Turn-Based Algorithm: each player sees the opponent's heroes from the previous day; if someone attacks, the turn-based mode reverts to the moment of the attack (the turn is interrupted for the player who was attacked, and the turn of the player who attacked is rolled back by one day); this will allow you to play with turn-based mode and Tournaments like 1v1v1v1 throughout the game.

Please, before making any suggestions about the turn-based mode, first understand how it works now (and how regular turns work in Heroes in general).
It must be understood that during the turn-based mode, the blue (synchronous) player's turn does not yet exist. It will only exist after the red (previous) player completes their turn.
In Heroes, players take turns, and this order cannot be violated; otherwise, how do you transition from turn-based mode to regular gameplay, and how do you end the game? How do you resolve conflicts? For example, both players capture the same neutral Castle in one turn; do you roll back both turns, and then what? Do they both capture the same Castle again? Or should someone graciously yield?
Are you, DaU, suggesting juggling the order of turns?
Therefore, currently, in case of a conflict (or contact), the blue player's turn (the turn that doesn't yet exist) is rolled back, and the red player continues as they were. Because the red player would be doing the same even without the turn-based mode.

An aspect that is not obvious to many, even experienced players:
During the turn-based mode, the blue player is behind the red player in terms of intelligence data about them. That is, during the turn-based mode, the blue player has information about the red player that is one day old (compared to regular gameplay).
And during the rollback due to contact or conflict, having the opportunity to replay the last turn, the blue player does not gain any advantage.
On the contrary, the ability to replay the last turn is not enough to compensate for the outdated information about the red player before the encounter. The blue player would still have made a different move in the replayed turn in regular mode, having up-to-date information about the red player. Moreover, having up-to-date information, the blue player would have made different moves in the previous turns leading up to the contact. Therefore, a more balanced option than the current one would be, for example, rolling back two turns for the blue player and one for the red player.
For the sake of balance, it could be suggested that the red player also be one day behind the blue player in terms of information about the opponent. But in games against AI, this is impossible (think about why).
At the same time (if we understand that the blue player is in the red player's past but in their own unconfirmed future), the blue player may receive intelligence that they would not have been able to obtain in regular gameplay, for example, scouting the red player's hero, who is actually already in a location that is inaccessible to the blue player's view or who is actually already without an army. But this will have a role-playing (enhancing the blue player) effect only if the red player is ahead of the blue player in the pace of scouting the area and is moving not towards the blue player but past or away from them. And it still won't be able to compensate for the blue player's overall lag in information from the red player before the contact. Those who disagree, write examples refuting these statements, even if they are hypothetical. The topic, as it turns out, is not as simple to understand as it may seem.

+ while writing this answer, I realized that with the current turn-based mode, if there are AI opponents between the players, for example, if one player is red and the other is green, then the green player will be able to move before the blue and beige AI opponents. Which shouldn't happen and will inevitably lead to "temporal paradoxes," violations of the natural logic of the game. Therefore, for games with the current turn-based mode, players must be in a sequence (blue-red / green-orange-purple / turquoise-pink, etc.).