Revealing mistake: In the beginning, there is a scene where Sen. Shaw is watching Raymond make a speech on a TV screen. It was obviously put in in post-production because the aide immediately next to the screen is nodding and clapping in agreement at inappropiate times.
Revealing mistake: When Raymond Shaw's skull is being drilled to implant the chip, bone "dust" comes out. This never happens, since always some blood comes out and the bone debris are mixed with blood and therefore unable to "fly" like wood or plaster debris.
Revealing mistake: Sen. Jordan is supposed to be an expert kayaker, but when paddling his kayak it is very obvious from the way he splashes the tips in the water that he cannot paddle.
Chosen answer: The tattooed woman was something that they all dreamt about. She was significant to the capture and the tattoos stood out among all of the other arabs.
Toolio
I always felt they were just there to cause confusion. If they started to remember the image of the Arab woman/ women would sound and look crazy trying to explain. However later in the movie when they shoulder flash backs the women were not there and Shaw choked out that other guy but, early the Arab women where there, and he had plastic on his face choking him out. So, all the memories are mixed up.