Run score?
 

Men's slow pitch.

Bases loaded, 2 outs.

Batter hits a ground ball to the right of F6 who fields it cleanly. Since his momentum is going toward 3rd base, he throws the ball to F5 who is moving to get to the base.

R2 assumes he will be out by 15 feet and overruns third base towards his dugout. F5 cannot control the throw from F6 and drops the ball.

Yelling ensues: from the offense telling their runner to get back to the base; from the defense telling F5 to tag the runner.

F5 has retrieved the ball and manages to tag R2 before he can get back to the bag.

R1 has crossed home plate prior to the tag of R2.

Does his run count?

I think you are describing what was originally a force out.

You said "overruns third base". Was that a "reached third" or not?
Reaching third removes the force.
But, where was R2 when tagged? If still past third, then it is not a force, so the run scores.
The way it would have been a force was if R2 never "reached" third or retreated to be between second and third.

If the overrun of third was toward the right side dugout, maybe R2 did not pass third.
Please clarify.

To clarify:

R2 overran 3rd base and did not touch the base. His dugout was on the third base side of the field. When tagged, he was in foul ground and on the home plate side of the base.

Your question is about force or timing play for the third out.

If I understand your play, R2 reached 3rd to remove the force; then was tagged out on the overrun.
("F5 has retrieved the ball and manages to tag R2 before he can get back to the bag.")

As long as R2 does not return to precede third base, which would reinstate the force, I see it as a timing play.

So, "R1 has crossed home plate prior to the tag of R2" means the R1 scores.

The issue is judging whether R2 actually reached 3rd, and i'm reading this as yes.

When tagged, the runner had passed the base, so no force. HOWEVER, if there was an appeal for the missed base as a 4th out appeal before the infielders leave the infield, that WOULD be a force out.

That would not be possible in USA Softball, because the 4th out appeal isn't being made on a runner that has scored; but would be valid anywhere else, I believe.