Appealing for a Fourth Out?

 
With one out and runners on second base (R2) and third base (R3), the batter hits a line drive down the right field line. Both R2 and R3 believe that the ball will drop in for a single, so both advance to their next base. However, the right fielder makes a spectacular catch for the second out.
 
Neither runner understands that the ball was caught, so with R2 standing on third base and while R3 has already crossed home, the defense throws to second base to appeal that R2 never properly tagged up. The umpire agrees and calls R2 out for the third out. The defensive coach then immediately asks that umpire to also declare R3 out for not properly tagging up.
 
The offensive coach argues that the defense cannot appeal for another out after the third out has been recorded. The defense coach wants the run taken off the board because R3 did not properly tag up. He also reminds the umpire that R2 was forced out at second base, so no runs score when the inning ends on a force out.
 
What do the umpire do now?
 
Call
The defense may appeal that a runner left a base on a caught fly ball before the ball is first touched (NF 8-2-6a). The appeal can be during a live ball or when the ball is dead. In the situation below, the defense properly appealed that R2 left second base before the fly ball was caught, so R2 is out for the third out of the inning. Since the defense also immediately properly appealed that R3 also left early, R3's run does not score.