How can I get the cloudy film off of my clear plates?

If you can find it a product called Zud works for that

It is caused by hard water, I'll leave it to others on the glasses, but get a water softener to stop the problem. Yes, they use salt to regenerate.

Soak in vinegar, then wash and dry by hand.

Try soaking them in vinegar water. It should help loosen the mineral deposit. It can happen even with soft water, but worse with hard water. It may take more than one try. If it doesn't come off, the glass may be etched and it will never come off, same goes with glasses of any kind. I normally hand wash clear glass, unless it is already etched. I have never had a problem with clear jars yet and I have hard water. The last three years is the first time since the nineties that I have actually had a dishwasher, so hand washing dishes is no problem for me.

I have well water and had this issue when we first moved into our house. I now add citric acid (you can buy it on Amazon.com), to the compartment next to where I put my dish detergent. Fill up the compartment (about 1-2 tablespoons). As soon as I started doing this, the film was completely gone.

Use vinegar.

Sibyl Daniels, I agree, this is the best site ever for encouragement for whatever you are thinking about or actually doing anything.

Nancy Turner, we have been hand washing for most of fifty years of marriage. I would use the dishwasher, if it worked, but hubby does 99% of the dishes, so I don’t get a vote! ☺️

If you have been useing a dishwasher to clean them, they will need to be proffessionally re-polished. Not really worth doing, cheaper to by new!