May I ask why you don't seal the tile? You can seal with a matte finish as well as a gloss. Keeping the floor clean would be so much easier if sealed.

you don't need to seal a slate floor necessarily but you do need to seal the grout for sure, you can mix 1 Tbsp of borax and 1 Tbsp of liquid castile soap I like Dr bonners peppermint now you must disolve the borax in boiling water then add castile soap and a couple of litres of plain hot tap water. This will not damage the stone the grout or the sealer. Do it in sections sloshing a layer of mixture over floor and push around a little with a rag scrubbing any dark grout and leave for 5 to 10 minutes. Now mop up and move on to next section. This sounds like a lot but you really should only have to wash these floor every few months if you don't have heavy dirt around just keep them well vacumed and spot washed If you have any chips that show white you can rub a little vaselin on them to blend it in. You should check if your installer did use a sealer and have it sealed a sealer won;t make or shouldn't make it shiny and these are truly carefree low maintence floors I love them they don't show the dirt easily.