Who's on third?

[Carl Childress]

R1, R2, 0 or 1 out. Fly ball to right, and F9 busts toward the line to make the catch.

Base umpire (BU) must get the retouch of both runners. Right?
Plate umpire (PU) has catch/no catch and fair/foul. Right?

If R1 retouches and tries for second, BU has the play.

If R2 retouches and tries for third....

Who has the play?

I have always taught that the BU splits the difference and prepares for plays at either (or both) bases.

A supervisor for an independent minor league says the PU should take that play at third, even if it's the first play after the catch.

Any thoughts?