Yes, I have finally seen that documentary and know about the mic in his tie so I assume that's how he recorded his solo numbers. That process is straightforward enough so that someone like me (who knows absolutely nothing about filming and recording musical numbers in a movie) can understand the entire process.
Here's what I don't understand. In The Rain in Spain number, do Harrison and Hyde-White sing live? Does Hepburn also speak her lines leading up to the song live and then just lip-sync Eliza's lyrics so that Nixon's vocals can be added later? Or is Nixon actually on the set (but out of sight) and singing the lyrics to Hepburn's miming? I know these questions must sound stupid to those who understand the whole process, but honestly I just want to take this opportunity to find out exactly how the whole recording thing is actually done.
BTW I think you have done a great service to all of us non-insiders (I hope I'm not the only one) by bringing up all these issues regarding the My Fair Lady film. I'm really learning a lot of things I never knew or really understood. Even though I have owned the DVD of My Fair Lady for at least 15 years, I never bothered to watch the documentary (on disc #2) about the making of the film until yesterday.