There are steam cleaners that extract water, but they are intended for carpeting. And they are very pricey. The concept of steam cleaning is that very little water is used, so there is no need to suck the water back up. You may be looking for something like the Hoover H3060 FloorMate SpinScrub 800. It sounds wonderful, but I don't think it gets good reviews from homeowners. For one thing, it doesn't heat the water that goes into the machine. Is your grout sealed? If you haven't applied a sealer to your gorup, then it likely is not sealed, since builders and installers don't do it. Here is a link that might be useful: Hoover Floormate SpinScrub

Oh, yes, I also tried Zud which contains oxalic acid. In all cases I soaked the surface for 3-4 hours and for some cleaners I did overnight. So my feeling is that acids either do not work or very strong acids are needed where tile damage is likely. Given that this is a rental, I am looking for more gentle approaches.

It looks like the wall has deteriorated behind the tile, which isn't unusual when water has had places to get behind the tile, especially if the apartment was built many decades ago. What is peeling off is probably a silicone caulk that someone put over the grout lines to attempt to reduce the water infiltration through the grout lines, is my guess. My experience with my old house (built in 1929) is that there is no fix for the tile after there is so much water deterioration of the wall behind the tile, short of removing all of the damaged wall, replacing it with cement board or whatever is the most up-to-date wall product these days, and then re-tiling. I'm sure you're not going to do that work for a place you rent. What we did before we bit the bullet and replaced the tub/shower walls, is to neatly duct tape white vinyl shower curtain liners to the wall to prevent further damage. Put them up high on the wall but make sure they cover all the way down to the top of the tub. The liners are cheap---like $3-$5. They keep the wall dry. And you won't have to look at the crappy walls if you get white liners rather than clear. It's not an elegant fix, but it will work for 3 months. In any case, old wet walls deteriorate and it shouldn't be something that the landlord holds you responsible for. Not sure how to clean it for now. Elbow grease, scrubby sponges, and household cleaner with hot water? Looks like it will take some doing.

Thank you all for your great ideas and for taking the time to reply. I went with Angela's idea though. My mother in law was up this weekend and she mentioned the Arm and Hammer stuff too. And then my sister in law took me to the dollar store and there it was! She bought me some for $2.25. We brought it home and sprayed it on and let it sit for just a few minutes and everything came off! The water was GROSS! Ug! But the bath tub is very white and CLEAN now. Thanks again! :) Happy cleaning, and may you all have lovely WHITE bath tubs! Haha!