Black tea develops cloudy dregs; how to avoid? I'm not much of a tea connoisseur*. But I do know what I like &mdash; strong, sweet black tea, and lots of it &mdash; and at present, the method I'm using is to boil a cup or so of cold water and about nine cheap tea bags, then turn off the heat and let the whole thing slowly cool on the burner (usually enough that I don't need ice to bring the result to a slightly warm drinking temperature). Then I pour it in a pitcher with some sugar and enough water to bring it to half a gallon; I generally squeeze the bags a bit to avoid waste, although I've tried avoiding that a few times to see if it makes a difference. This tastes nice and stout, and gets me through the average day. I've read some things that suggest steeping tea this long and aggressively will make it horribly bitter, but either those are exaggerations or I have a barbaric tongue, because I haven't noticed any such problems.

The problem I do have is that, especially after a bit of refrigeration, the last half-cup or so of tea has a lot of nasty-looking and nasty-tasting particulates in it. They're fine enough it's difficult to settle them out, even trying various slow pouring methods without disturbing the pitcher much, so presumably the bags don't filter them out either. What can I do about them other than throwing out the dregs? Is there some flawed part of my process that's producing them?

*To say the least.
Those particles are bits of tea leaves (etc.) that came out of the tea bags. They're fine enough particles to get through the bags. Basically, tea dust.

You don't notice them at first because they're suspended in the tea. So you could just stir it up before pouring off each cup. Alternatively, disturb the pitcher as minimally as possible, and pour off the tea. Leave the settled tea dust at the bottom.

Rinsing the tea bags in cold water before steeping might help. (Make sure to use cold water so you don't remove much flavor.) 

Other than that, a fine enough filter will remove them. You could try coffee filters, they're cheap enough. Or a nut milk bag. Or a superbag. (Coffee filter is probably the finest filter of those, though by far not the sturdiest).

PS: You might be covering up the oversteeped tea flavor with enough sugar...