In a dirty Thompson, the woman does a handstand on the floor with her ass sticking in the air and her legs spread. the man puts his balls in her ass and his dick in her pussy, or vice versa. when he 'finishes' he pulls his balls out of her ass (or pussy) and there is a popping sound. the man usually then says "Booyah!" or something similar and slaps the woman's ass like a cowboy.
(makes cowboy noises and gestures) When i pulled 'em out, i said "giddy-up!"