Regarding your Theatre of Mind statement
If a combatant uses a thick tree trunk as ¾ cover (+5 AC) and an opponent moves to attack — it seems natural when playing Theater of Mind that a defender interposes the tree between them and the first attacker.
I would like to note that, as you mention, all turns are taken in parallel during a round, even if the system itself is turn-based. I could swear this is stated explicitly somewhere in the PHB but I can't find it right now. Anyway, to make it completely clear, this is about how the scene can be imagined, not how the game works.
This section discusses the Theater of Mind imagination, not the combat system itself. I wrote this part so OP could understand that actually there is no incoherence between the turn-based situation and the dynamic situation, as long as you keep it consistent.
While yes, it means that you could move around while he tries to attack you, it also means that you were just attacking (or doing whatever action you did on your turn) and you are now more worried about defending yourself and making sure you are aware of your surroundings (i.e. there is no other enemy at your back) than doing footwork around the tree.
For example, if both of the characters involved in this combat were archers shooting bows, you could think this, instead, as if when you decided to attack him, he also attacked you back almost at the same time. Or that while you were running away, almost getting to behind the tree, he was already moving to get a better angle and shooting, so you have no time to do your footwork around the tree any more. The "doing footwork" would be using your action to Hide or Dodge, maybe (depends on your DM, might be Ready as well), then you would be fully concentrated on your footwork and not doing other actions.
If you are thinking on a melee situation, well, try to imagine it not as a
- I run behind the tree
- The enemy approaches the tree
- I start moving around the tree
- I run behind the tree
- The enemy is chasing me the whole time
- You never had cover to begin with.
The main incoherence here seems to be that you are mixing the "turn-based" feeling from the grid when migrating your imagination to the Theatre of Mind. Either everything is turn-based or everything is dynamic. If you make yourself dynamic on enemy's turn, but he is not dynamic in yours, then you will indeed get conflicted.
The Combat System
RAW, you can only move on your turn, as well defined in the Chapter 9: Combat from PHB, unless you Readied some action that allows you to move or you have some kind of Reaction that does the same, so you would be behind the trunk in your turn, and on enemy's turn, if he can get a better line of sight to hit you, you lose the cover (or get only half cover), so...
No, he is not moving to interpose the tree, unless he uses his action (or reaction) to do it somehow and the DM allows it.
Note that mechanics that allow you to do something as a reaction are specific, and I don't know any specific mechanic that allows you to do what you are suggesting.
From the comments:
So does that mean in Theater of Mind an attacker can always use their movement to maneuver around the tree and remove the cover advantage just like one can do on a Battle Grid?
(Battle Grid is a more common terminology than Battle Map, from my experience)
Yes. In general, it depends on the distance between attacker and target, since maybe the attacker can't get line of sight, but the attacker should be able to do it. But there are a few more points to consider
- The attacker might not be able to move sideways, either because it would trigger an Attack of Opportunity from a melee combatant or because the terrain doesn't let him.
- If the target is trying to gain distance, making the attacker move sideways instead of in his direction is an advantage.