I shaded parts of it (from pics I think if you're looking at what would be his right hand side) using vallejo model air armor brown glazes which really muted the green and made a nice shadow. On the other side (his left hand side) I used vallejo model color dark sea blue glazes. It started out with me just wanting to play around with different shadow colors to see how they compared on the model. No other real reason as to why. I was sort of in the same boat as you and wanted to play around and learn different effects I could achieve playing with saturation and colors of my lime green marine. So I pretended maybe he was in an environment where from different facings he was experiencing different colorations of light that would affect the shadows. I'd say just go buck wild and try something similarly fun!
I'm no expert but when I want something to become less saturated I usually try to add a more muted shadow color to the base color or add the complementary (in this case the reddish armor brown) to the green desaturates it. White and black also desaturate but change the brightness of the color as well but if you're shading then you'd want darker and could add a little bit of dark grey or black to desaturate and darken for shadows.
Hopefully those with a better understanding can chime in and correct me where I'm wrong and give further advice.