When any part of the hull crosses the finish line. (Not the bowsprit, nor nor spinnaker, nor crew, etc.)
It is often said that as long as the line judge treats all boats the same, then it all works out OK. But that is not true.
Consider a race with a 35' boat and a 45' boat with crossing the finish line determining the winner (forget PHRF for a minute). Both skippers are sharp and the hull of each boat crosses the start line at the same time. At the finish line, the hull of each boat crosses the finish line at the same time. Who wins?
If the line judge uses the rule of any part of the hull, then the race is a tie. What if the judge uses the stern? (As in, the entire boat has to cross to finish.) The 35' boat wins as the stern of the 45' will cross some time later. If both boats are moving at 4 kts, the 45' boat finishes 2.7 seconds after the 35' boat.
If a boat crosses the starting line when any part of her hull crosses, then the only fair way to judge her finish is when any part of her hull crosses (normally her bow).
The basic principle is the same even if using PHRF. For two boats with the same PHRF, the shorter boat will win if both boats cross at the same time and the stern is used to call the finish. For different PHRF values, the math gets more complicated, though it works out the same: The shorter boat is favored using the stern to call the finish. As the "crossing" point moves forward, the difference in favoring is lessened, until any part of the hull crossing the line is used for the finish.
That, I believe, is why RRS says a boat finishes when any part of her hull crosses the finish line.