Horrible Mechanics

[PeteBooth]

The main problem with your OP is this

Batter hits a bloop into center field, which F8 charges and appeared to trap. BU gives no signal and no verbal.

No one knows what is going on. If the BU gave no signal it appears he has NO CLUE along with the Players and YOU on what's going on.

Each of us has our OWN responsibilities out there.

As both TEE and Rich pointed out with R1/R2 the PU covers third base on the tag UP UNLESS it's a CLEAR CUT Single which from the OP it was not.

Here is what has me puzzled

Apparently BU ruled the ball was caught because F8 threw the ball to F3 to double off R1.

You say APPARENTLY BU Ruled. What's that! Either we have a catch or no catch. We do not have a MAYBE.

Based upon the STRICT wording of your OP it appears NOTHING was called because of your phrase APPARENTLY BU ruled. Therefore, EXACTLY what happened on this play? Was a call on the catch / no catch EVER made?

If R1 was doubled off with an apparent no call on the catch / no catch where was the OM because surely he is going to get tossed.

IMO, once we KNOW what happened then perhaps your question about what to do can be answered more effectively.

Pete Booth