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Combat is characterized in the game as a series of feints, parries, and footwork until finally a character gets an opportunity to attack. The Initiative systems are proposed as an abstraction of those emerging opportunities from fluid combat.

The take away is that combatants are constantly in motion.

The conflict comes when using Theater of Mind versus a Battle Map.

In Theater of Mind - a PC might say "I am using the tree as cover" - and it is assumed that the PC will interpose the tree when an attack comes - constantly moving around the tree to keep it between them and the attacker.

On a Battle Map, however, the PC places their figure - and an attacker can virtually always move to one side of the tree or the other - thus the PC almost never gets cover from a tree on a Battle Map.

Should we assume that a PC using a tree trunk on a Battle Map does or doesn't move to interpose the tree like in Theater of Mind? What are the guidelines around combatants interposing a tree trunk between them and the first opponent to attack them in Theater of Mind versus Battle Maps? Does that guidance change when a second opponent attacks?

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The Cover sections says

[...] A target can benefit from cover only when an attack or other effect originates on the opposite side of the cover.

If you attack from behind the tree trunk, then the target gets a cover bonus. However, if you decide to move around the trunk and attack, then the target doesn't get the cover bonus.

Keep in mind that a round lasts about 6 seconds. During that time, you can do fancy footwork behind the tree, but only when you do the attack, the DM decides if the target receive or does not receive the cover bonus depending on how you attack.

Answer this question:

When you do the attack, does the target is behind an obstacle? If the answer is yes, then the target receives cover bonus.

This is true even if you use a melee attack, reach or not reach. The obstacle might be small enough to not occupy a square (on a battle grid), but obstructive enough to grant cover bonus to anyone hiding behind it.

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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.

The Imagination

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

but as

  • 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.
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  • \$\begingroup\$ 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 Map? (i.e. the PC can never use a tree as defensive cover unless the opponent is stationary) \$\endgroup\$ – Praxiteles Apr 20 '18 at 3:51
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    \$\begingroup\$ @Praxiteles Yes. Actually, it depends on the distance as well (the attacker might not be able to move enough to get a line of sight if they are too far away), but generally, yes. I can add some suggestions on mechanics to how a party can use the Cover mechanic a little better than that, though, if you wish. \$\endgroup\$ – HellSaint Apr 20 '18 at 3:54
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The creature can't move off-turn for free to interpose the tree

This is true without a battle map as well. When you play in Theatre of the Mind style, what's abstracted is the grid, not the combat system. You still take turns, and the turn-based nature of combat is still so ingrained into how everything works that stuff won't really function at all like it was intended if you try to let people act semi-simultaneously.

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