Who gives the signal?
 

A14 is driving the lane. Stops, travels, gets fouled.
Simultaneous whistles: T has the travel, L has the foul. L begins walking to the table and giving the numbers when T intercepts to say the travel occurred first.
Who should give the signal for the travel and the throw-in spot?
When a calling official receives new information from another referee (i.e., a tipped balll OOB) the calling official changes his call and gives the signals, but we weren't sure in this situation.

Doesn't sound to me as if anyone "changed" their call. You just decided to go with the travel because it came first.

The T reports the travel and we get the game moving along right away.

The T should probably report the travel. It's not going to look pretty either way, so don't worry so much about it. When you do announce it, though, make sure you state loudly the travel happened first; especially if everyone knows the foul was about to be called.
Again, it won't look pretty no matter what you do, but getting it right won't always look pretty.

First of all the L should have recognized there was another whistle. The T should have come running in letting it be known they had something different.

The bottom line is the T called the travel, not the L. So the T should give the signal, but at the very least the T should be saying to their partner, "I see what you have, but the travel was first." Then you get the ball back in play and move on.

Peace

Yes, he should have, however:

It's not odd to have both whistles go at the same time, and the L not realizing there was a T whistle due to sound alone. If the T had now lowered his open hand signal, to walk to the L, the L could still be in the dark as to another call.

Which he did.

It might be a good idea for the T to stick with the play and should still see this after-travel contact. S/he should then confirm with the L that the contact he is calling is the same after-travel contact the T observed. This mechanic prevents cases where the T missed contact that the L saw. Granted, this is not needed in all cases, however.

I have seen calls botched because the T assumed contact that the L called to be same contact that he saw.

This is why you do not just lower you hand. ;)

He did not do it in the right manner based on what I read. He allowed his partner to go to the table. Now either his partner was so clueless or he did not make sure his partner saw him originally. Also you have to know to some extent there is a possiblity that your partner might have something. Not always, but most of the time for sure.

What I am talking about takes a second or two. I am not talking about a long drawn out conversation. You likely should also talk about these kinds of situations in the pre-game. Just a short conversation about this tells you and your partner(s) how you will handle this.

True, but I do not think this is that kind of situation.

Peace

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