How to determine Tower shield cover against an adjacent, Large creature?

Question

Suppose that a Medium creature stands 10 ft. away from a Large one (e.g., after taking a 5 ft. step back), and uses a Tower shield to grant itself total cover towards the direction of the Large creature.

The Large creature then uses a 5 ft. step to approach. It would attempt to approach diagonally, to attack around the shield (using a melee weapon, with 10 ft. reach, as normal for a Large, tall creature). The creature uses the rule for Big Creatures and Cover to select the most advantageous square:

Any creature with a space larger than 5 feet (1 square) determines cover against melee attacks slightly differently than smaller creatures do. Such a creature can choose any square that it occupies to determine if an opponent has cover against its melee attacks. Similarly, when making a melee attack against such a creature, you can pick any of the squares it occupies to determine if it has cover against you.

As far as I can tell, for any movement other than down-left, the Tower shield will grant total cover without further clarification.

In this situation, does the Medium creature have cover from the attack?

The relevant cover rule for cover is the following one:

When making a melee attack against a target that isn’t adjacent to you (such as with a reach weapon), use the rules for determining cover from ranged attacks.

I see two possible interpretations of how the reach and big creatures rules might interact:

1. The Large creature is adjacent to the Medium creature and making a melee attack, so it must take the least advantageous corner of its chosen square (top right), which crosses the Tower shield edge and grants total cover.
2. The chosen square of the Large creature is not adjacent to the Medium creature, so it may take the most advantageous corner of its square (bottom right), which does not cross the Tower shield edge.

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Note: This question is similar to this one, but specifically addresses Large creatures and larger.

The cover rules, specifically regarding the difference between melee and ranged attacks, for reference:

To determine whether your target has cover from your ranged attack, choose a corner of your square. If any line from this corner to any corner of the target’s square passes through a square or border that blocks line of effect or provides cover, or through a square occupied by a creature, the target has cover (+4 to AC).

When making a melee attack against an adjacent target, your target has cover if any line from any corner of your square to the target’s square goes through a wall (including a low wall). When making a melee attack against a target that isn’t adjacent to you (such as with a reach weapon), use the rules for determining cover from ranged attacks.

Answer 1

The medium creature has no cover in your second illustration, only in your first

The rule tells you:

Such a creature can choose any square that it occupies to determine if an opponent has cover against its melee attacks.

So, to determine cover you can chose a square. This must also count for resolving adjacency in regards to cover; otherwise chosing the square would not fully work to determine if the opponent has cover, which it says it does.

This also makes logical sense. In this situation, the target is not adjacent to the source of the attack, as there is a 5-foot gap between the squares. While the creature may be adjacent, the source of the attacks is not, and it makes sense using the rules for adjacency that relate to attacks coming from that square similar to when you attack with reach from further away.

Therefore, you can use the rules for ranged attacks for your melee attack, as quoted above by you. These read:

To determine whether your target has cover from your ranged attack, choose a corner of your square. If any line from this corner to any corner of the target’s square passes through a square or border that blocks line of effect or provides cover, or through a square occupied by a creature, the target has cover (+4 to AC).

The best corners you can pick are either of the two lower ones. For both of them, if you draw lines to the two upper corners of the tower-shield wielder's square. These lines run along that border, does this count as passing "through a border"?

The rules unfortunately do not clarify this, and there are lengthy discussions about the question if passing through a corner (and in extension through a border alongside) counts as passing through the border. The prevailing opinion, and your own assumption given in option 2 is no. While the rules text does not speak about this, there is a graphic in the core rules that supports this interpretation, as it shows that a line passing through a corner is not blocked:

We also always played it as not granting cover in those cases, as there is really no shield or obstacle between the creature in that square and the attacker. It is just along one of the sides. And with everyone understanding that, and it being used for both PCs and monsters, we never had an issue playing it like that.

Obviously, it is not total cover, as in the second illustration, you can draw several lines, for example from the lower corners of the attackers square, to the lower corners of the target, that do not come anywhere near the border occupied by the tower shield. Even in the first illustration you can draw a line from the lower left border of the large creature to the lower left corner of the shielded creature that is not crossing the line of the tower shield, so it will not have total cover.

Total Cover: If you don’t have line of effect to your target (that is, you cannot draw any line from your square to your target’s square without crossing a solid barrier), he is considered to have total cover from you. You can’t make an attack against a target that has total cover.

Tower Shield's "total cover"

Tower shield's description says:

That edge is treated as a solid wall for attacks targeting you only. You gain total cover for attacks that pass through this edge and no cover for attacks that do not pass through this edge

A situation where you can draw lines to the space that do not go through the edge of the tower shield, is by the defintion of total cover above, not total cover. I think what the language tries to express is that attacks that go through the shield side cannot affect the creature, there is not merely an increased AC.

All that being said, the rules could be clearer and more explicit here, so it may be best to check with your DM how they handle this.