As stated above, P class is the 3-class FC cabin designator for the routes between SFO-JFK/LAX-JFK.
As compared to the domestic 3-Class FC, the main difference is the food and quality of service.
(As stated above also, this usually means a serving of soup in addition to the regular offerings and the sorbet.) (Although I've gotten soup from HNL-SFO on the B744 redeye a couple of years back.)
I've flown in P Class during the period from 1996 through 1998. I could remember one flight on a B777 in FC, we had a nice Breakfast service. Numerous types of Biscotti to start. I got a selection of my main entree. (Got to choose from some omelet, or apple crepes.) Then they came with a breakfast cart that had some freshly baked (or they just seemed freshly baked) pastries, croissants, etc. along with some sausages, bacon, and filet mignon. (Yummy! This was one of the best steaks I've ever had on a plane) To finish, the FA's offered Godiva chocolates.
We also got one of those "hand held" video units and got to choose from tapes that they had on board. (The B777 PTV's were disabled on this flight. Besides, the old FC PTV's were not capable of playing tapes) It was nice to have these units, but I just didn't know where to place it if I had to go to the bathroom or when I was finished viewing the movie. (The FA's would just ask which movie you wanted and they'll set up the unit for you.)
It was nice. Wonder how it is lately....
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SFnFlaGuy