What acid-free cement stain remover can you recommend for cleaning freshly laid tile?
While clumps of mortar can be beaten off, the "veil" is known to be only a thin dried film. It is alkaline in nature and therefore can only be removed with acidic cleaners.
If you are afraid of chemistry, you should stay away from COLA, for example. A litmus paper dipped in the drink shows a pH value of 1 (highly acidic).
So (this is serious) take a bottle of COLA, pour it over the affected area, wait about 10 minutes, and then pick up the dirt with a rag. Wipe with clean water and you'll see: the cement film is gone!