How can I remove really tough soap scum from tinted glass shower doors?

I'd try vinegar first with a scrubbing sponge. And then I'd go over it with baking soda on a damp sponge.

There is two things you can use and they both worked you can use grease lightin or awesome.

You can try pure vinegar and if that doesn't work go to a hardware store and get something called STP. It is a commercial cleaner.

Once you have as much of the scum off as possible. Use a sponge to apply cooking oil to the inside of the doors (I do not recommend Olive Oil--rather a vegetable based oil). This not only hides the existing scum but proves to inhibit additional build-up. A Military Housewife's trick to passing the inspection test. It works, Wonders!

Try Scrubbing Bubbles. You can buy it in any store. You'll find it with the other cleaning supplies. I am not positive of the company that makes it, but it is called Scrubbing Bubbles. I have had really good luck with it.

VIM liquid cleaner works like a charm on soap scum both on doors and showers/tubs themselves.

Just apply, leave a couple of minutes, regular sponge with a little elbow grease!

Property managers and plumbers use Edfred cleaners because time is money, and they usually have the very worst jobs.

There is an Edfred step by step with plenty of photos of removing rust, soap scum and lime from a shower door here http://www.edfredcorp.com/blog/?page_id=200

If you google the name you can read user reviews and other photos.