I’ve got a Woods PRD8400 and an RD60 rear finish, rear discharge mowers. Both are living in a barn on the floor on the wheels (not standing upright) in a shady spot in the corner. I just finished maintenance on both of them and I never have any trouble with milky oil in the gear box now. Even not after 10 years of the same oil in the gearbox.
Here is a story for you: when I first got the rear discharge mower, one of them always gets a bit muddy. Being nice as I thought I was, I cleaned the deck top and bottom with the garden hose and it the pressure washer. Even so I didn’t directly spray on the gearbox, the oil was milky in the gearbox and the bearings in the spindles were bad faster.
I quit cleaning the mowers, leave them dirty and just lube the spindles after the 3rd use. Sharpen the blade and check the oil every year. After 10 years, the oil gets replaced and I’ve got no more problems with the oil being milky.
The other problem is that it takes a while for the milky oil to be totally replaced. I think I had to fill the gearbox up twice or three times before the milky oil disappeared. I can imagine a cold day and then the sun beating on it makes the gearbox breath. So find a cool spot, no sun, no water = no trouble.