ONE faucet in the house emits some "black, seemingly greasy/oily goo, most of the time (not always??), when my wife runs water in the tub.
It does NOT come from the shower, as far as I've ever been able to tell, (which I always take showers) of the same faucet...it's when running a bath only, and it is the ONLY faucet in the house that does this.
It makes a greasy ring around the tub...Clorox bathroom cleaner "melts it right away" to scrub off with a Scotch pad.
It's a Delta one handle faucet which gradually mixes the water, so it seems it isn't specifically hot or cold...it's just there.
Any ideas? So far it's totally isolated to this one faucet, and the tub only...been doing it for quite some time now???
Suggested source or origin? Any hope of a fix other than replacing the faucet and piping back to the source?

IF it is the faucet, then replacing the core should clear it up. IF not, then there is no way for us to diagnose it.

A possibility is a galvanized nipple possibly inside the tub spout. They rust and can cause this to happen. To eliminate this remove the spout take a look see.

This can occur with older braided steel flex lines. The inner rubber liner of some flex connectors will starts to deteriorate. If you have flexes connecting your water heater, I would looks at those first.

The tub spout will flow more water than most anything else in your house, and you /might/ find some black deposits in your aerators.

I had the same thing in my house. I opened the wall behind the valve and the valve was connected with the old style braided lines. I piped it correctly and the black went away.