It's called Crystal Clear.
It's made by a company called, of all things, Aquarium Products.
One wonders how they came up with these bizarre names. :-)
It took five double-strength applications over the course of a week, but now the water in the tank is crystal clear. Really. I kid you not. As I've mentioned before, bettas don't care how clear the water is, they come from muddy and polluted ponds and breathe air from the surface. But I like a pretty tank, and for the same reason the cloudy water doesn't bother them, the clarifier chemicals don't either.
I haven't tried it, but I suppose they could live okay in untreated tap water. Not sure how much nasty chemistry they would absorb through osmosis, though. And I don't really want to intentionally kill a fish to find out.
In other tank news, I got tired of the bare tank, and could see the fish were pretty bored with it too, so I put the big ornamental rocks and the castle ornament in the top rack of the dish washer and got the algae off, and put them back into the tank. It gives the fish some places to explore, and hide. I'll take a picture one of these days.