Black Spots on underside of Awning - How to Remove?

My wife just successfully cleaned ALL the black spots off. Tilex did do the trick for us. She did the underside of the awning in small sections, spraying the Tilex on, then scrubbing it, then rinse off with water. Then move to the next section, repeat process.

Thanks to all for all the great suggestions! Be careful with the Tilex if you use it -- it seems to be strong, so wear old clothes plus my wife wore eye protection when working with it over her head.