The TAFM might get you in the ballpark.
Very hard to say what was used for fuzz or how it was recorded when going by a studio recording.
But what it DOES sound a bit like to me is the old Shin-Ei Companion Fuzz. It has that same "velcro-like" fuzz tone. The originals are very rare and expensive but there are some clones available like this one
or this one
which even looks like the Shin-Ei.
The Shin-Ei Companion fuzz can also be heard on these two Radiohead tracks: Myxomatosis Exit Music (For a Film)
The Vinyl Thief fuzz (very cool track BTW) doesn't seem quite as thick in the low end.
Another option that I can think of that would probably work to get you close would be the Dr. Scientist Frazz Dazzler, one of the new ones with a blend, like this one
Here's a demo. Check out the tone around 1:05 when he's using it as a gated fuzz. http://youtu.be/kZwpV-yVgVc
Of the pedals I currently own I think the Chunk Brown Dog would get me closest to that tone with the gate up fairly high, some clean blended in, fuzz rolled toward the synthy side and on the "hard" setting.
Hope that helps.