What do you do when this happens...
 

Sounds like a, "don't overthink it, this would never actually happen" scenario, but it actually did happen this weekend. Middle school game, A1 shoots jump shot from the corner. After he has completely landed, B1 runs up to him and gives him a hard shove with his body to gain position for potential rebound (clearly a foul, but nothing that warranted intentional foul). As soon as I whistled the foul, A1's try for goal became lodged in between rim and background. The foul and the lodging happened near to simultaneously, if not simultaneous then so close together that there's no way to tell what happened first. My partner and I decided to report the foul, and then give possession based on alternating arrow. Would anyone do anything differently?

Yes. B committed a common foul. How should you resolve that?

I'm inclined to think the ball lodging was just a missed shot and since B had a foul, it would be a team A throw in at spot nearest foul.

The first half of that is true. But, you did.

You need to decide what happened first, and enforce that. You can't "enforce both," which is what you did.

What BlueDevil said.
Bob J also.

Hard body box out foul or a lodged ball with an Alternate Possession.
Foul every time it's, protection and sportsmanship.

Thinking about it slightly differently, what if it was a dribbler who got body bumped at approximately the same time they traveled? Would you enforce both, report the foul, and go AP? No way. Decide which came first and go from there. If it's close, I'm going with the foul.

Interestingly, this could be considered a case where the whistle makes the ball dead as there is a degree of judgement in determining that the ball is lodged. I've seen balls thought to be lodged slowly become "unlodged" following the whistle prior to any attempts to manually dislodge it. To my memory, there isn't any defined period for how long the ball has to remain in the lodged position, so the situation requires the judgement of an official to declare it lodged. If you whistled for the foul before you or anyone else could whistle for the lodged ball, I think we know which you judged to come first.

From that point, the proper enforcement has already been discussed.

Whether the ball was stuck or not, the shot was missed. So unless a teammate of the shooter got the rebound (aka "got player/team control") before the foul was called then we have a common foul against B1.

Would this make a difference?

Sorry, for some reason for part of my reply I was thinking a player from team A fouled.

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I just wanted to make sure I wasn't missing something. :)