Well, here's an interesting combination.

The War Caster feat says:

When a hostile creature's movement provokes an opportunity attack from you, you can use your reaction to cast a spell at the creature, rather than making an opportunity attack. The spell must have a casting time of 1 action and must target only that creature.

The current text of Illusionist's Bracers says:

While wearing the bracers, whenever you cast a cantrip, you can use a bonus action on the same turn to cast that cantrip a second time.

In the Player's Handbook - Combat section, it says this about bonus actions:

Various class features, spells, and other abilities let you take an additional action on your turn called a bonus action. The Cunning Action feature, for example, allows a rogue to take a bonus action. You can take a bonus action only when a special ability, spell, or other feature of the game states that you can do something as a bonus action.

The "on your turn" could be describing what various class features, etc. do, not laying down the law that it has to be on your turn.

The next paragraph says:

You can take only one bonus action on your turn, so you must choose which bonus action to use when you have more than one available.

But again, this is focusing on the limitation of only having one bonus action on your turn.

More importantly, remember from the Player's Handbook Introduction - Specific Beats General rule:

If a specific rule contradicts a general rule, the specific rule wins.

So read Illusionist's Bracers again, and it sure looks like this:

1. On a creature's turn, the creature provokes an opportunity attack from a war caster with Illusionist's Bracers.
2. The war caster casts a cantrip at the creature as a reaction.
3. The Illusionist's Bracers grants a bonus action to the war caster that can be used on that same turn.
4. The war caster casts the cantrip again with that bonus action, not on the war caster's turn.

This should be fixed by saying "on your turn" in Illusionist's Bracers, or the PHB introduction should say explicitly, "A bonus action can only be used on your turn."

Well, here's an interesting combination.

The War Caster feat says:

When a hostile creature's movement provokes an opportunity attack from you, you can use your reaction to cast a spell at the creature, rather than making an opportunity attack. The spell must have a casting time of 1 action and must target only that creature.

The current text of Illusionist's Bracers says:

While wearing the bracers, whenever you cast a cantrip, you can use a bonus action on the same turn to cast that cantrip a second time.

In the Player's Handbook - Combat section, it says this about bonus actions:

Various class features, spells, and other abilities let you take an additional action on your turn called a bonus action. The Cunning Action feature, for example, allows a rogue to take a bonus action. You can take a bonus action only when a special ability, spell, or other feature of the game states that you can do something as a bonus action.

The "on your turn" could be describing what various class features, etc. do, not laying down the law that it has to be on your turn.

The next paragraph says:

You can take only one bonus action on your turn, so you must choose which bonus action to use when you have more than one available.

But again, this is focusing on the limitation of only having one bonus action on your turn.

More importantly, remember from the Player's Handbook Introduction - Specific Beats General rule:

If a specific rule contradicts a general rule, the specific rule wins.

So read Illusionist's Bracers again, and it sure looks like this:

1. On a creature's turn, the creature provokes an opportunity attack from a war caster with Illusionist's Bracers.
2. The war caster casts a cantrip at the creature as a reaction.
3. The Illusionist's Bracers grants a bonus action to the war caster that can be used on that same turn.
4. The war caster casts the cantrip again with that bonus action, not on the war caster's turn.

This should be fixed by saying "on your turn" in Illusionist's Bracers, or the PHB should say explicitly, "A bonus action can only be used on your turn."