I'm guessing CLR is like Lime-Away. You are fighting mineral deposits, maybe with some soap residue blended in, which are hard to get rid of when they build up enough to be seen that much. I have lived in areas where the mineral content in the water is pretty bad so understand your challenge. I have used Lime-Away with success, but don't recommend it as a first line of defense.

Find a good organic cleaner (7th Generation Sun and Earth, and EarthWise, and vinegar water come to mind) and a green scrubby, and use elbow grease to scrub those doors clean. Anything that will eat through that film without any elbow grease is really bad for humans and animals. Go with something a bit gentler that requires some effort on your part, and work at it until it's de-filmed.

Let us know what you used and how well it worked for you.

Once you have scrubbed those shower doors, don't forget to use a product such as "Rainex" to keep them clean! Water will just bead up and the doors can be cleaned with a very mild cleaner. Redo every few months according to use.

I also use Rainex to keep my shower doors steam-free. There is a product line just for that. I apply it quarterly and can see the results, as it were.