Are monks only allowed to use non-unarmed weapons as off-hand when two-weapon fighting?

Do the special rules regarding monks’ unarmed strikes prevent monks from using unarmed strikes as off-hand weapons when using two-weapon fighting?

The monk class’ unarmed strike section states:

There is no such thing as an off-hand attack for a monk striking unarmed.

The context of this is dealing with 1x vs. 1/2x strength, but as a side effect, does it mean a monk must use a weapon rather than an unarmed strike when using two weapon fighting to generate more attacks per round?

Two-weapon fighting states:

If you wield a second weapon in your off hand, you can get one extra attack per round with that weapon.

It then states later on, confirming that unarmed strikes may normally be used:

(An unarmed strike is always considered light.)