When Hubby worked in the US VI, I bought a beautiful bamboo mat. After one warm season on our porch, it had mildewed and gotten "ugly."
I washed it well with bleach and soap, taped off the fabric borders, and went to work with a big paintbrush and "chicken house" blue primer (chicken-house-blue b/c that is what color we painted our chicken house and 2 gallons of paint later...there are A LOT of things around here that got painted chicken-house-blue as well).
I love stripes, so I carefully taped lines at random intervals and painted different colors with various paint that we have on hand.
You can see where the paint bled a bit under the tape where I got heavy-handed with the paint application, so that would be my only words of caution....do light coats of paint as opposed to heavy.
Not perfect..but better than the mildew-stained look.