Actually there is an official video, but it is not in any way remotely SFW and can only be found on the official MA website. It is the song layered over an interview with the woman who starred in The Devil In Miss Jones, and there is a ton of hardcore clips from the movie.
Not entirely from "The Fall", there definitely some Brothers Quay mixed in (either "Street of Crocodiles" or "Rehearsals For Extinct Anatomies").
I think I've given up hoping for anything that quite matches Mezzanine (one of my top 10 albums of the last 20 years)... But I'm glad they're still doing work.
The clips are entirely from The Fall, and the stop-motion sequence was created by Wolfgang and Christoph Lauenstein. I thought it was the Brothers Quay, too, though.