Fair/foul balls are not usually subject to arguments. There are always exceptions though. Here is a play in a recent Yankee game.

What does it take to be a fair ball?
A FAIR BALL is a batted ball that settles on fair ground between home and first base, or between home and third base, or that is on or over fair territory when bounding to the outfield past first or third base, or that touches first, second or third base, or that first falls on fair territory on or beyond first base or third base, or that, while on or over fair territory touches the person of an umpire or player, or that, while over fair territory, passes out of the playing field in flight.
I added the emphasis. When the ball is bounding it can be over fair territory and still be fair. It does not have to be touch fair territory on either side of the base. First and third (well all the bases) are in fair territory.
Put it all together and a ball that bounces over a base is a fair ball. This is what happens in this play. The ball does not land fair after passing the base – but it does not have to. Good call here.
A few inside baseball things here. Notice how the field umpire made this call. This is normal procedure. Up to the base is the home plate guys call and anything that bounces over is taken by the field guy. The Yankee manager first argues with the home plate umpire who correctly tells him he did not make the call.
On the last replay, you see the field umpire call fair ball. The home plate guy is looking at him and vigorously shaking his “yes”. I think he is trying to help his partner out in case he was blocked.
All in all a good call on a tricky play. Page 37 of RuleGraphics contains fair/foul ball. See examples of the book and information on where to buy at our website.