In a poodle grooming book it says that the reddish brown stains are caused by tears which oxidize. I've had 4 white (actually cream) poodles over the last 39 years and never had the problem. I clean the eye-boogers whenever I see them accumulate. Maybe that prevents irritation and tears?
When I was young, I admired clever people. Now that I am old, I admire kind people. -Abraham Joshua Heschel, theology professor (1907-1972)