I'd bet there's a simple option for treating them, laquer, varnishes, etc...whatever it is, though, it'll soak up alot of it.  Maybe linseed oil applied with a (old) paint sprayer?  Could get an even coat with air, just then you'll have overspray.

We had some old barn siding & bats that we left outside in a rain (accidentally), didn't take long for the undersides to get covered with fuzzy mold.