Something I Didn't Catch The First Time Watching This
Actually, the first time watching something a work should just be allowed to sweep over you.
However, on second viewing I noticed how consistent Julee Cruise voice sounded no matter whether she was on the stage or hanging from the rafters.
The natural conclusion was that it was dubbed. Either she just lipsynced/sang to a pre-recorded version (my guess) or post-dubbed the music to the live performance. In the interview on the DVD, Angelo Badalamenti mentions that they were happy with Julee Cruise in the studio session, which gives more credibility to it being pre-recorded music.
You can also see - particularly when she is hanging in the air - her lips not syncing to the words (at times).
Also, the video shots (at least some of them) were pre-recorded when Julee Cruise was singing while in the trunk of the car. Close-up shots/angles don't match. If you want to see this, notice the monitor and Julee in the background. Her head movements don't match (best seen @ 34:14 in when she sings "shadow in my house. Or 34:37 when she sings "I want you rockin' back inside my heart" where the camera angle on the monitor doesn't match where the camera operator is standing.
Sort of a pre-cursor in a way to Mulholland Dr. when lip syncing is revealed in Club Silencio. "It is all an illusion."
Living Is Easy With Eyes Closed