The term "Italian tile" isn't enough info to know the best way..Natural stone, fired, glass?
CLR uses acid which can etch stone and title. And it's effect on grout can be unpredictable. Whether this tile etches or not or in a way you notice is kind of subjective. This issue will return, so more cleaning, more acid, more etching maybe. Certainly hard on the grout. So long term cleaning is a bit of an issue.Testing in one little spot might look OK but when the entire area is done you might notice a sheen change, maybe uneven.
And of course there could be minerals acids wouldn't remove.
I'd try Bar Keepers Friend. It isn't as abrasive as Comet type powers and it's an acid, but pretty weak. Combined with the little abrasive the little bit of acid is more effective. I don't even wear gloves, but you should. Pretty darn safe imo. Dirt cheap. It's about the only cleaner I use inside the house too.
You can bring out the big guns and use a needle scaler. Works great, pretty safe. Good at cleaning the grout. About $50 at Harbor Freight if you have a compressor. And who doesn't have a compressor?
Or call a swimming cleaning company. They're use whatever they think is best when they see the tile, and may use several things. Their experience, if they have any, is what you'd be depending on.
