Well tannins are chelating agents, and it looks like they bind relatively well with iron, and precipitate out of the solution.
Here is an experiment you can perform.
1. Get you're black tea (whole leaves)
2. Set up two pots of boiling water (or however you make your tea)
3. Brew two batches simultaneously for the NORMAL amount of time you brew your tea for...say 3-5 minutes?
4. Pour those two brews into separate containers and make sure you label them NORMAL-No Iron and NORMAL- Iron
5. Repeat the brewing of 2 more batches, but this time brew them for longer than you normally wood, say 10-15 minutes. (The longer you brew the more tannins are released)
6. Pour those two batches out into separate containers label them LONG BREW-No Iron and LONG BREW-Iron
Now as a control you can place some tea leaves in some room temp water and put it in the sun or something, but I don't think its necessary
IMPORTANT!!! Make sure you use the SAME MASS of leaves in each of the brews, AND THE SAME AMOUNT OF WATER, so the mass of the tea is the same, as is the volume of water (to prevent dilution and varying concentrations).
NOW, you have 4 brews, 2 labelled no iron, 2 labelled iron.
I recommend finding an Iron supplement from the grocery store (tablet form) and then grind up the iron pill as best you can into a powder, and dump it into the two batches labelled IRON.
The reason behind this is that Iron atoms will chelate with the tannins, and form a solid precipitate and fall to the bottom of the cup. (theoretically)
I think you should try this while the tea is still warm (but not super hot as that could interfere with chelation of the iron to tannin molecule)
Then, pour all 4 teas through coffee filters, even the one with no iron added.
Then taste test them and have someone else do it too who doesn't drink a lot of tea like you do.
The one's WITH IRON added should THEORETICALLY be less bitter. The more bitter, the more tannins you have. But if the iron takes them out, then yo uhave a more concentrated tea without the bitterness.
good luck