Chrome on Shower Door is Discolored?

I have a chrome and glass shower door. The glass is fine but the chrome frame is slowly developing black streaks on the inside, growing downward toward the shower floor. I thought it might be mildew but bleach and other cleaners have no effect on it.

Also, mildew is unlikely on a metal surface and the streaks seem to be becoming almost part of the surface itself, as if the chrome finish is changing chemically somehow. Help! Any ideas on what I have and what to do?

Ted from McHenry, IL