Batter-Runner Tagged While Retreating

 
With two outs and a runner at third base (R3), the batter (B1) grounds a slow fair ground ball about half-way down the first base line. As the first baseman fields the grounder, B1 stops running, and B1 begins to retreat toward home plate. The first baseman chases B1 toward home plate and tags B1 for the third out of the inning. B1 is tagged just after R3 crosses home.
 
The offensive coach argues that R3's run should count since B1 was tagged after R3 touched home. The defensive coach argues that the batter is out the moment he begins to treat towards home, so the inning is over and no runs score.
 
Which coach is right? What does the umpire do now?
 
Call
A batter-runner may retreat toward home plate as long as he does leave his baseline, or touch/pass home plate (NF Casebook 8.1.1a). No runs score if the third out is made on the batter-runner before he touches first base (NF 9-1-1a). In the situation below, B1 legally retreated, but he was forced out before he touched first base, so no runs score on the play.