The rules are worded weirdly, and I will disagree with KRyan, and the FAQ, on their reading. Woe is me...

TL;DR: If the Monk chooses to use Two-Weapon Fighting, she may use her full Strength modifier with her off-hand unarmed strike.

First, we need to consider the regular rules. For Damage (PHB, p. 135):

Damage

If you score a hit, roll damage and deduct it from the target’s current hit points. Add your Strength modifier on damage rolls involving melee and thrown weapons. If you’re using a weapon in your off hand, add one-half your Strength modifier (if it’s a bonus). If you’re wielding a weapon with both hands, add one and a half times your Strength modifier (if it’s a bonus.)

For Two-Weapon Fighting (PHB, p. 160):

Two-Weapon Fighting

If you wield a second weapon in your off hand, you can get one extra attack per round with that weapon. [...] You can reduce these penalties in two ways:

• If your off-hand weapon is light, the penalties are reduced by 2 each. (An unarmed strike is always considered light.)
• The Two-Weapon Fighting feat lessens the primary hand penalty by 2, and the off-hand penalty by 6.

From this, we conclude that:

• If you wield a second weapon in your off hand, you can get one extra attack per round with that weapon.
• There are penalties for attacking with your off-hand, and you only get to add 1/2 your Strength modifier as bonus to damage.

As noted by KRyan's answer to the question you linked, there is the issue that technically one needs to wield a weapon in one's off hand to gain one extra attack, and one wishing to make an Unarmed Strike typically doesn't -- though the Monk could.

From the highlighted excerpt about an Unarmed Strike always being considered Light, which only makes sense if making an Unarmed Strike with the off-hand is possible in the first place, I interpret the rules as allowing using Two-Weapon Fighting without wielding a weapon in one's off-hand.

Then we need to consider the Monk's Unarmed Strike (PHB, p. 41):

Unarmed Strike: Monks are highly trained in fighting unarmed, giving them considerable advantages when doing so. [...] There is no such thing as an off-hand attack for a monk striking unarmed. A monk may thus apply her full Strength bonus on damage rolls for all her unarmed strikes.

Invoking the principle that context is everything, I will consider the two last sentences together.

In isolation, the first sentence does not make much sense. "striking unarmed" or "strike unarmed" appears nowhere else in the PHB. The proper game term would be "making an Unarmed Strike", which is used consistently.

By reading the first sentence in light of the second sentence, however, it all makes sense:

• The second sentence grants the Monk the benefit of using her full Strength modifier with all Unarmed Strikes; thereby overriding the Damage rule for off-hand damage in this case.
• The first sentence therefore introduces the second sentence: it's an opening line, in plain English. Just fluff, with no game consequences.

And in this light, the formulation of the first sentence makes sense:

• Plain English sentences are not restricted to using game terms, so "striking unarmed" is not an issue.
• When the Monk makes an Unarmed Strike with her off-hand, it deals the same amount of damage as a regular attack... as if it were a regular attack. As if it was not an off-hand attack. Hence, from an observer, "There is no such thing as an off-hand attack for a monk striking unarmed", all attacks are indistinguishable.

Hence, as a result:

• It is permissible to use Two-Weapon Fighting without wielding a weapon in your off-hand, in which case the extra attack is an Unarmed Strike.
• All Unarmed Strikes from a Monk apply the Monk's full Strength modifier to damage, as a special case overriding the regular 1/2 Strength modifier on off-hand attacks.

I must warn, though, that the result is less than stellar:

• Flurry of Blows gives: -2/-2 at the start, -1/-1 starting from 5th level, no penalty from 9th level, and an additional attack from 11th level.
• Two-Weapon Fighting gives: -4/-8 in the absence of the feat, and -2/-2 with it.

Furthermore, Flurry of Blows also grants the full Strength modifier for the "bonus" attacks even when attacking with Special Monk Weapons.

There is the Two-Weapon Feat fighting line, which for the cost of 3 feats would allow a Monk 15 (or higher) to make 3 more attacks, however it uses iterative attack penalties for the extra attacks, on top of the -2 penalty, so even then seems less attractive than only 2 extra attacks at full BAB. Especially on a BAB-starved class.