I have "Read" about a lot of supposed quality cleaners, but what would prevent a over proportioning of the chemicals from settling on the porous portions of a tile product creating a dulling to the finis.
Bleach, diluted to a 10% solution with 90% water is "Supposed To" be one recipe, but I would do test swatch sample cleanings in an inconspicuous area first, in the event that it bleaches the color out of the roof tiles.
There is a contractor that had previously posted on this forum, by the name, (I Think), of Apple Roof Cleaning, who was out of Florida, if I recollect properly.
I think he had some posts with pictures located in the photo section of the Roofing Sub-Forum over on www.ContractorTalk.com
I do know for a fact, that wherever he posted at, he did have links to his company website.
He may have had some products available for the handy DIY'er to purchase.
I do not know of him personally, but just from viewing his roof cleaning photos.