Looking for help with with vintage Milus watch and possible Omega case?

I purchased a womens vintage Milus watch at thrift store yesterday because I liked the band. I didn’t know much about Maker other than they were considered high quality and people like em. I didn’t know they were part of Tissot family - when I started going over the watch, turns out it is solid 18k and the only watch I have found with same band is a vintage gold Omega.

Is there anyone with a deeper knowledge of Milus watches, Tissot, their relationship with Omega and if it’s likely that Milus (or Omega) would have used Omega case and band to sell off remaining Milus stock when they started to slow down?

Any help much appreciated - it’s not a band style that you see very much so finding info on it hasn’t been easy.