Why does this tea turn cloudy when refrigerated?

This is caused by precipitation of the tannins in the tea. They are in solution at the higher temperatures, but as the tea cools they are forced out of solution. All of those little solid bits make the tea look cloudy.

The richer the tea is in tannins, the more likely it is to do this (e.g. high-quality black teas brewed for a long time). The peppermint tea doesn’t cloud because it lacks tannins.

Hard water exacerbates the problem, because it has a harder time holding tannins in solution. Rapid cooling is also more likely to provoke the precipitation.

Just to echo @thorninmud and include a link to Goodhousekeeping for ice tea. They include clearing the cloud the next day with adding boiling water.
My aunt use to make the ice tea the day before in the refrigerator, cold water and tea bags in a large glass pitcher.

Add a pinch of baking soda to the tea when it is first brewed. That will clear up the problem.

Interesting! Thanks so much for the answer… We will now be able to play around and see if we can keep it clear.

All black tea turns cloudy when put in the refrigerator. One of the national brands used to advertise that it never turned cloudy. It changes the taste to me. I will not drink it after it goes into the refrigerator. So, my suggestion is to try not to brew more than you need in a day and never put it in the refrigerator.