First, do the same check on the water you have going into the tank - from the hose, etc. That way you can determine whether a "whole house" filter is needed if it is coming in from outside. Bleach will kill any microorganisms growing in the water - bacteria, algae, etc. But it won't clean it out. That dead stuff, if not flushed out, can settle to the bottom of the tank, and then later be picked up and show up later. So after bleaching, fill and empty your tank several times - normally you won't have to flush that much, but if you have a build up....The good news is that stuff usually flushes out pretty easily.
Or, it may be lime scale - minerals that have precipitated out of the water into the tank. Most of it will stick together, just like it does on shower walls at home, or on coffee makers, if you have naturally hard water. But some of it can be knocked loose and show up as floating bits. It is a lot "stickier", and is harder to clean out.
While not the most powerful cleaner, the safest for that would be white vinegar. A couple of gallons in the fresh tank, circulate through your system, and let it sit at least several hours, or even days. Then again, drain and flush. Don't, however, bleach and use vinegar at the same time.
The good news - while gross, none of the above (if you've bleached previously) is harmful, and is actually a bit nutritious! But if you've tried both of the above, and are still seeing things, you may want to think about a PuR water filter on the kitchen faucet for drinking/cooking use, keep on flushing the tank by normal use (ie - fill the tank, and skip the city water connection) as it could take a while to get it out.
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