I can imagine that the mildew is on the mylar side of the dacron. Looks difficult. I don't have any other great ideas to remove it though. Nice benifit of living in a dry climate!
I did a Google search and found allot of references. Here is one...
Sail Care (From J World - Ullman) Dirt or caked salt. Use a soft bristled brush and liquid detergent. Avoid hard powder detergents and stiff brushes as they may damage the finish and stitching.
Oil, grease, tar, and wax. Use warm water, soap and elbow grease. Hard stains can be removed with household bleach or common stain remover. Be careful to thoroughly remove all cleaning solvents or they will damage the finish.
Blood. Soak the stained portion for 10-20 minutes in a solution of 10 parts water to 1 part bleach. Scrub and repeat if necessary. Rinse thoroughly.
Rust and metal stains. Scrub with soap and water, then apply acetone, M.E.K. or alcohol. Rinse thoroughly and dry.
Mildew. Hot soapy water is usually all that is needed. If necessary, use the diluted bleach approach. If a residual chlorine smell is still present after rinsing, a 1% solution of sodium thiosulfate will remove all chlorine traces.
Paint and varnish. Acetone and M.E.K. should remove most common paint stains. Varnish can easily be removed with alcohol.
Use all solvents with care. Always rinse and dry thoroughly. Solvents and bleaches can damage nylon and dacron if not removed properly.
Mylar and Kevlar sails can generally be taken care of in the same manner as their dacron counterparts. Avoid solvents whenever possible as they can break down the adhesives used to laminate the nylon to the substrate.