Yeah that's true. Similar thinking to what I had in mind with Tilex. I may try a small spot and see.
I spend a lot of time in Seattle-we get pretty good at removing mildew here in the Pacific Northwest. I've used bleach (Clorox) for years on colored canvas table umbrellas that are not nearly as stout as the military covers. I remember reading the Tilex label and thinking bleach was the main active ingredient.
The bleach solution doesn't have to be very strong to remove the mold/mildew and the mild solution doesn't do much damage to the fabric or colors.
Like Suprman recommends, I use a pretty soft brush to wet down an area and let it soak for just a few minutes, then when I come back it doesn't take much scrubbing at all.
Usually rinse with a hose to finish & remove bleach (and find the spots I've missed).