The main cause of roof stains depends on the climate, location of the building. As mentioned, you are in central Florida with a hot and humid climate is the main cause of this. One of my friends also had a similar issue with his slate-coloured shingled roof who is lives in the north of Canada. The humid climate creates algae and fungal growth in shingles which make these stains. He has tried both chlorine and oxygen bleach for this stain removal. Even though chlorine bleach is the most effective one, I would recommend oxygen bleach considering its effect on nature.