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

As a bugbear:

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

Using a reach weapon:

"Reach. This weapon adds 5 feet to your reach when you attack with it. This property also determines your reach for opportunity attacks with a reach weapon."

I am trying to determine the cut-off ranges for various situations. 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).

But how do I calculate opportunity attacks?

"You can make an opportunity attack when a hostile creature that you can see moves out of your reach."

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'.

So an opponent can be next to the character and safely back away 5' because they have not left the 10'/15' reach. But what happens at 10' and 15'?

At 10', they are still within my normal reach, but only on my turn. At 15' they are leaving my reach, but I only get that extra 5' on my turn so it would not be available as a reaction.

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

Generally, you make your opportunity attack when a creature 10' away from you tries to move farther away from you.

You have quoted most of the relevant text. When it is not your turn, your reach is 10'. The Long-Limbed feature is no longer a factor. So just like any other character, you make the opportunity attack with your polearm when a creature that is 10' away from tries to move further away.

The only exception would be...

If you are making an opportunity attack on your turn, you make the attack when a creature 15' away from you tries to move farther away from you.

The circumstances for this will be rare. If, for example, you are a College of Valor Bard who uses a polearm and you cast Dissonant Whispers on your turn, targeting a creature in melee range with you. The creature fails its save and uses its reaction to run from you. Because it is still your turn, the opportunity attack against the creature is triggered when it tries to leave your 15' reach.

15 ft on your turn, 10 ft on other's turn

Long Limbed is active on your turn, making your threat range effectively 15 ft. You can make opportunity attack against enemy leaving 15 ft on your turn, or 10 ft on other's turn.

Though rare, enemy can provoke opportunity attack on your turn, for example by casting dissonant whisper while they are next to you.

Use the reach you have at the moment

You do not have a "base" or "normal" reach as far as the mechanics are concerned. At any point in time, you have a given reach with a given weapon, though. With your halberd, your reach on your turn is 15" and otherwise 10". Opportunity attacks are triggered when an enemy leaves your current reach.