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If you see it in all faucets:
When hydrants are flushed (our town does annually in the overnight hours), the high volume and pressure clean out our old iron water mains like a steel brush would, sending rust and loosened deposits to the end of the line where the hydrant is. Although a lot of that water is run out of the hydrant, the remaining sediment in the line will always run to the hydrant for a few days afterward, and the area homes will get the discolored water. This is also true when there is a water main break (many in the cold northeast with old water systems), and some sediment from the soil around the broken pipe could get in as well. We also saw this problem when the town reversed flow direction for some neighborhoods while repairing gate valves in the mains.
If it is just the tub/ shower:
I know when changing a toilet flapper(after we tampered with it) it will sometimes leave bleed black. Perhaps one of the washers in your faucet(if it has black washers) is bad. Does it have any problems turning off? I doubt it has anything to do with the water heater if its only that faucet and not others that is causing the problem. If it is hot side only, it is probably the washers within the hot side of the faucet. They are deteriorating. Hot water is harder on them then the cold side. Remove the valve and replace it with a new valve insert. Do not forget to replace the valve seat within the valve housing. If you do, you are wasting your time. Replace both hot and cold sides.
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