Garage Roof - dirt ingress

Chickereeeee

I have an old garage with a clay-tile hung roof. The tiles are hung on (wide) battens, visible from below, with about 25mm gaps between. Through the gaps, the tiles are visible. I had some work done on the garage (including re-tiling) just after we moved in, but  did not get a permeable membrane fitted. The garage was clean and dry, after about 70 years, so didn't think about it!

Now, after a windy day, I get all sorts of dirt showering down from  the inside of the roof. If I leave something in the garage unmoved for a few months, it is covered in a layer of dirt.

Has anybody come up with a solution for this problem, which doesn't involve re-roofing (again)?

Thanks