Go industrial strength on it. Have you ever worked in a restaurant or someplace that had one of those big yellow mop buckets on wheels with the mop ringer built into it? You really need one. They can swab the deck like nobody’s business. It might seem extreme, but with all that tile there is no alternative. You will go through those cheap squeezy mops like Kleenex and they will never do 1/4 as well. If they were not fast and efficient then every restaurant in the world would not have one. You can get them for about $70 at Costco or other such places. You will not regret it. Just fill’er up and slosh that mop water around freely, then suck it back up and ring it out over and over. if that doesn’t get your floors clean enough to eat off of then your feet must be excreting black schmoe or something. And do not underestimate how much crud you drag in on your shoes. If you are used to carpeting then you should know it sucks up a lot of dirt. Nasty. But every little pebble you carry in on your feet in a tiled house is just going to lie there waiting for you to step on it again. So leave your shoes at the door. It sounds like a pain at first, but thats just what you do with tile or wood. And it will always be cleaner than any carpeted house.

how old of a house? what kind of tile? how clean/ drafty is the building envelope? fireplaces? old windows? what type of hvac(furnace)? the black is coming from somewhere. superclean a 10×10 area and walk it. any black? no? cover it. wait. walk on cover. any black? if it still blacks your feet after the initial superclean, your cleaning lacks cleanliness or your tiles are creating the discoloration. also remember your feet wiil pick up everything because of the oils they excrete. ask a carpet guy if they recommend walking barefoot or in socks on carpet.