get the solution Jay mentioned. Take a coffee can of the solution and a tooth brush. Work on one area the size of a silver dolar and see what happens.
Bleach and water will work, water and vinegar (white) will work, light mixture of TSP will work, and even 90 proof alcohol will work.
Two things cause stains on comp roofs: they are mold and mildew and foreign material, iron, in the components of the grit applied to the shingle fabric.
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