A few weeks ago I was listening to a radio talk show and a gentleman in the car business called in responding to a woman with the same frustration.  He said to use automobile buffing compound and fine steel wool as instructed on the can to successfully clean  shower doors.  I just tried it and I am excited to report it does a great j0b.  After all these years, having suffered with multiple chemical sensitivity in the past, I find this is very low odor and seemingly non-toxic.  Within  30 t0 45 minutes my shower was sparkling clean looking almost new.  It also cleaned the brass and floors as well.

You will need:

Rubbing Compound
Gloves
Fine steel wool
A pale of water
Rags

Instructions are to clean a small area at a time with the steel wool as an applicator.  Do not allow the compound to dry.  Instead, once applied and rubbed for desired amount of time,  wipe it off with a damp cloth.  You will want to rinse out your rags as you go.  Change the steel wool as you need but do not get it wet.

A word of caution;  do not use any other clean in conjunction with the rubbing compound.   Spraying the shower with a bleach or mildew cleaner,  for example,  will result in toxic fumes.