Hi, and thanks in advance for your help.
Please let me give you all the facts, then the description.

I live in a 50 year old apartment building in Brooklyn, NY.
My family has been this apartments only tenant, the entire time.

Though not a "luxury," building, tenants are paying from $2,000 to $4,000 for apartments here.
So it's supposed to be a nice place...

Suddenly, just in the last few weeks, the bathtub is backing up every morning, leaving a nasty residue rising up to six inches around the tub.
It backs up even if I close the external valve, (pictured).

I have to spend twenty minutes scrubbing this nasty residue, because it is so extremely difficult to clean off.
(Doesn't everyone spend the first twenty minutes of every day scrubbing their tub so they can then shower and catch the 7:15 am express bus?!)

The handymen here have snaked it, and it makes no difference.

I am including pictures of this detritus, which as you can see is reddish, and sticks stubbornly to the tubs surface.

What's worse is I have a new roommate moving in for September, and I imagine if this is not cleared up promptly, I could lose the tenant...

Has anyone ever seen this sort of debris before, and can
1. identify it's possible causes, therefore possible solutions?
2. identify what this debris is, or might be?

As snaking the line was totally ineffective, could it be the drains, air vent is clogged? (I'm assuming drains have air vents...)

Thanks again for any and all your help, and suggestions.
This is becoming a huge problem, and now may very well cost me, not only real money, but worse losing a roommate I have spent a lot of time interviewing for.
Heck, I wouldn't move in if the tub looked like that every morning.

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