I have been laying floor tiles for the past few days and have been
religiously cleaning them as I go.

However, even though I am using clean water, buy the time I have got
the cement off (there is quite a bit to clean, as I have been making a
mess), the sponge is full of 'grey' water.

No matter how hard I clean them, as soon as the tiles dry, they have a
milky white surface on them, which I guess is a thin film of dry
cement/adhesive.

There must be something which is good for cleaning them, but I guess
that as I am still in the 'dirty' stage, there is no point.

Once everything is finished and the floor grouted, what can I use to
bring the tiles up to an 'as new' finish?