Rule 6, Section 4
ART. 4 . . . The direction of the possession arrow is reversed immediately after
an alternating-possession throw-in ends. An alternating-possession throw-in
ends when the throw-in ends as in 4-42-5.
ART. 5 . . . The opportunity to make an alternating-possession throw-in is lost
if the throw-in team violates. If either team fouls during an alternating-possession
throw-in, it does not cause the throw-in team to lose the possession arrow.

Rule 4, Section 42
ART. 5 . . . The throw-in ends when:
a. The passed ball touches or is legally touched by another player inbounds.
b. The passed ball touches or is touched by another player out of bounds,
except as in 7-5-7.
c. The throw-in team commits a throw-in violation.

So what is the best instruction to give to the table crew about when to switch the arrow? If you tell them, "Don't switch the arrow until the ball is touched by a player on the floor", those instructions incorrectly cover this case. Is, "Don't switch the arrow until the ball is touched by a player on the floor or unless the thrower throws it away", too complicated? Is there something better?

I tell them to wait until the throw-in ends. If there is an unusual situation which would either retain or switch the arrow, then I let them know which way to have it pointing before we proceed any further.

Agreed.