A steam mop will dislodge dirt and hygienically clean the tiles.

Using a little vinegar or methylated spirits in the mop bucket should help.

Maybe get a professional carpet/floor cleaner in, their steam cleaning may work. Thin about where feet are before hitting the floor, perhaps it is the outdoor area doing the damage?

If they are standard floor tiles, using a little vinegar in the water you’re mopping with will help clean and bring out a shine in the tiles.

yeah steam mops are great and leave the floors shiny

Have you got a steam mop? They are quick to use and really efficient, maybe once you have make a rule no shoes in the house and must wear socks?