Putting carbon in would definitely clear it up... but it would also remove a lot of the nutrients that you have from the water. Activated carbon works by indiscriminately binding tons of stuff in the water, and so while the tannins would get bound up, so would the ferts.
MTS probably have little to do with the problem, though if you have tons of them, that may suggest that you're overfeeding the tank a little, which allows their population to explode.
I'm not entirely sure what the best method to deal with the tannins are, but my guess would be that if you used some really strong filtration WITH carbon for a short period of time, you could clear the water and then start dosing again. if you're dosing EI, this'll fit in nicely....
So basically, you might fertilize your tank on Monday and do a big water change on Sunday, or something. Instead of doing a big water change, you could throw activated carbon in your filter (and maybe even add another big filter with carbon) on Saturday, let it run until Sunday, and then restore your tank to how it was, do the water change, and then resume dosing as usual.
You may need to repeat this cycle a few times, since the tannins leach over time.
Also, I've never purposely tried to remove tannins, nor do I dose heavily, so I'm just speculating.