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I have crafted a Barbarian bugbear that uses a halberd weapon, and will take Polearm Mastery as a feat.

Bugbears have the Long-Limbed trait:

When you make a melee attack on your turn, your reach for it is 5 feet greater than normal.

The description of the Reach weapon property says:

This weapon adds 5 feet to your reach when you attack with it, as well as when determining your reach for opportunity attacks with it.

The second benefit of the Polearm Master feat (PHB, p. 168) says:

While you are wielding a glaive, halberd, pike, quarterstaff, or spear, other creatures provoke an opportunity attack from you when they enter your reach.

Making a standard melee attack is pretty straight-forward. He can attack anyone in 15' (5' for normal, plus 5' for bugbear, then add 5' for reach).

At what distance can he attack someone coming towards him?

So my reach is generally 15'. But per the bugbear description, that extra 5' is only when I make an attack on your turn. So his effective reach is only 10'.

Per Polearm Master, I can take an Opportunity Attack when they enter my reach; which is normally 15'. But I only get that extra 5' on my turn, and since this is another creature using it's movement, it's not the character's turn.

So can he attack when someone is 15' feet away, or does he have to wait until they are within 10'?

P.S. There is a sister question about Polearm Mastery and leaving my threat range.

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    \$\begingroup\$ You know it's the Bugbear's turn when his arms get 5 feet longer all of a sudden \$\endgroup\$ – user47897 Mar 11 at 19:11
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Long-limbed increases your reach on your turn. Thus, on your turn, your reach is 15. On anyone else's turn, your reach is 10. Polearm Mastery's triggered OAs follow suit - 15 on your turn, 10 on everyone else's.

It is possible to get OAs on your own turn, though relatively rare. In your case, if this happens, it will most likely be because you're fighting an enemy who can use their reaction to move on your turn.

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