[Tdol]

I don't see why the action has to have been done to qualify. I agree with your example being wrong, but with the backshifting examples, I can see a difference:

A It may rain today.
B (reporting the same day) He said it may rain today.
The next day
He said it might rain yesterday.

Here, I see the possibility described as having finished.

Also, I personally don't subscribe to the definition of tense being used there, as I don't see tense as simply a temporal relationship, but within that restriction, I still see 'might' as describing a completed past possibility of rain.