Thank you for that info on bleach. I had not known that.
I went for the white vinegar, as I've heard that when you use it to clean windows with newspaper, it gains better results than the apple cider vinegar.
I took a new sponge, soaked it with the WV, applied to the tank and waited a little bit. Then I used soft dry papertowels and buffed it. It came beautifully clean except for the very top edges of the tank, all the way around. Apparently, the previous owners were real pigs when it came to cleaning.
Then I thought..... I have apple cider vinegar with mother. I wonder if that, since it's essentially "live", will work better. Hey, it gets rid of toenail fungus, so it was a decent bet. So, I applied that to the top 2 inches all the way around. Alas, it brought the cloudiness back as it was before. That was completely unexpected.
I've just dried that off and reapplied the WV. Will go back downstairs and buff again. No trip to the gym today for bicep curls. This is gonna take care of that.