I have a hoover floormate that can be used on tile floors and it works great- it scrubs and sucks up the water- and is much more effective than mopping. You can use their solution or a vinegar solution- you cannot use pine-sol in it.

It sounds like the clay is absorbing any type of spill or moisture from sock feet. I would not try to sand it and it would probably come out quite disastrous.

If it were me and replacing the tile were not an option or desire, I would call in a professional floor company. Have them recommend a cleaning method or maybe they could possibly apply several coats of industrial wax and buff the finish.

I also have the hoover floormate and recommend it as well.. It really scrubs and cleans the floors well, but I wouldnt take it out to get rid of foot prints every time some one walked on the floors . a micro fiber mop may be a good solution for quick clean ups in-between.
I would stay away from pine-sol, it might be the residue from the cleaner left behind on the floor.. I would use vinegar and water and see how they are after removing the pine-sol residue .. This could very well be the problem here is a link to Bona microfiber mops, you can also use it on tile and they do have a tile cleaner as well.
good luck and let us know how you do!
Ree

I run a cleaning company and we use Shark Steam Mop on our jobs. It uses hot steam to clean, no chemicals. Works like a regular mop but you don't have to rinse or anything. It comes with machine-washable pads. You can also steam your carpet with it.
Also found this link that seems to be talking about your type of tile floor (see below

hmmm.. interesting, I never knew quarry tile needed that much maintenance. You learn something everyday.. !

Thank you for your responses, I do appreciate your suggestions.

My tile is glazed so I don't think it would qualify as quarry tile, would it? Truthfully I've been tempted to paint over it. I accidentally dripped a little paint on it when I was painting the walls and I never did get it all off. I was thinking about maybe doing a quilt block design on it at one time, and I also thought about just doing a marbelized or a rag faux finish on it. Only problem was I couldn't decide what colors to use so that if any of the paint wore off, it wouldn't look too bad for some of the original finish to show through. It'd be nice to tone it down some so it wasn't so.... red.... It's hard to decorate the rest of the room without it looking like Pizza Hut or a Coca-Cola ad.

I kind of thought Reeree might be right in that there might be an oily residue being left on the floor, which attracts dust and pet fur and makes the footprints show up so badly. To make matters worse, we have a dog that thinks he belongs in the house. I'm not a dog person, we got the dog for the grandkids who lived with us, and they took very little interest in him though they wanted him very badly at first. Now the grandkids are grown and the dog has bonded with DH. He (the dog LOL) tracks stuff in, and waits to scratch and shake himself till he gets inside. The fur in the house drives me crazy. I just can't keep up with it!

I would love to have one of those steam cleaners. I looked at both styles on Amazon.com and read all the reviews people had submitted on them. I do have a carpet cleaner that sprays out water and then sucks it back up. It's kind of big and cumbersome, but I tried it on the tile. I haven't used it since we removed all the carpet, so I had to reacquaint myself, but I just wanted to see if it would do 'in a pinch'. It did vacuum up most of the water but still left the floor more wet than dry. No puddles, though, and the water that I threw out was dirty. It has only a brush that spirals around a roller in the front and I'm not sure that exerted very much pressure, because of the machine being made for carpet. Still it was a lot easier to use than I thought it would be. I tried vinegar and hot water this time, but I soon noticed that the floor was drying out kind of cloudy. So I got down on my hands and knees with a towel to dry it off and then I saw that it didn't really get the floor as clean as it looked from 'standing up distance'. I'm 62 and there's a lot I can't really do anymore due to a bad back. I read all those comments in another thread about the old people whose houses are so filthy and they don't know it. I just don't want to be one of them but my main activity is my garden and my quilting.

I'm kind of wondering about the pads that the Shark has. How long do they last before you have to replace them? Does the Hoover floormate use pads or does it have brushes? Can you who have them tell me if there's anything you don't like about your machines and whether there's a specific model I should get? You mention steam, but I don't see in the stuff I can find on the Internet whether the machine heats the water or if you have to start with hot water, as I did with the carpet cleaner. This isn't exactly steam, if so. What is the hottest the water should be?

Don't buy a hoover Floormate. I bought one for about $175 and it only worked for about three weeks. It quit picking up the dirty water. Not fixable. Bad choice. JMH Experience.

I have the Hoover FloorMate wide path model. I also had one of the older models and picked it up at a yard sale for next to nothing and it worked great- I didn't know you couldn't use pine-sol in it and it eventually ruined it. It worked great though so I bought the wide path model- I only use vinegar or the hoover floormate solution.

Jannie-if after only three weeks the store should be more than happy to exchange or refund the money- not to mention it is still under warranty from Hoover and they would replace or repair it. There could be a clog in the suction part- I had that happen once and it wasn't picking up dirty water. I think someone on this forum actually pointed it out as the possible problem and how to fix it.