Who watches the runner?
 

I come to this site for guidance, so could use some expert advice and experience here:

2 Umpires working:
1 out; R1 on 3B; Batter hits a blooper fly ball to short RF; both F4 and F9 are running to make the catch.

BU is in position "D" but is drawn towards RF to see catch / no-catch. Now, who should watch R1 to make sure she doesn't leave 3B early? And how best to get the right angle to see both R1 and catch?

All tags at 3B belong to the PU, unless the BU is savvy enough to call him off in this instance. BU should NOT be drawn to RF, because catch/no catch in RF is never a BU call starting in the C2 position. But, BU would have to pick up BR if the ball drops.

PU should ease down the 3rd base line towards the holding zone, so that the tagging runner is visible in peripheral vision (unless the issue of fair/foul becomes a higher priority). At a minimum, peek, and see if runner is back, or heading back; once first touch, and as soon as can look away from the catch, get a solid look at the runner's position.

As BU, unless fairly sure that ball will drop, I would verbalize "I have your tag", and make sure PU hears, as well as R1 and the third base coach. I have found they don't leave early when they hear you announce that. If/when BU accepts that variation, PU can step up more toward the catch, without regard to the angle of the tagging runner. PU can get a better look at the catch, and even give help at 1B if the ball drops; in the play you describe, not very likely a close play at home if ball caught, but, PU, can return to 3rd base extended for that, if necessary, and there is no play at home if ball drops.

BU, starting from C2, then has a minimal adjustment to see R1 tag (or not), and can work straight toward 2B if the ball drops, knowing PU can now get out for the throw back to 1B, since he didn't have to go away from that play to get the tag angle.

It may sound complicated, but to an experienced crew, it just happens.

If the BU is going out to watch the catch the PU has to watch the tag. You probably won't see both the catch and the tag at the same time and you can't move too far up the line if you're expecting a play at the plate. You can try to listen for the ball hitting the glove and watching the runner or cheat a bit and watch the ball hit the glove and then look at the runner. If you watch for the catch and then look for the tag make sure the runner has clearly left early before honoring an appeal.

Fly ball and tag-up responsibilities should be covered in your pre-game conference with your partner. Don't assume you know who's responsible for what. Talk it over.