Try this product should work, test it on a small area first

I recommend, that you go to a tile shop and ask there what to clean it with. Some cleaners can damage the finish. The tile shop should also sell good cleaners for tumbled stone.

Thank you!

Doubt if the products shown would do much cleaning or dulling. The original power was a great product--subsequent manifestations not so much.

Maybe a steam cleaner? it shouldn't hurt either the stone or grout. I have this little one and it was not too expensive and works great on my tile counters (and many other things).

https://www.steamfast.com/shop/steam-cleaners/sf-275-canister-steam-cleaner

Stonecare International (SCI) has a product line for natural stone that worked well on our travertine floors, granite counters, and tumbled stone showers. It will keep the mildew away and not damage the stone. We bought it at a local vaccum and cleaning supply store.