Grammar question

I need professional opinions from my fellow grammar policepersons out there:

What is the plural posessive form of the word "lady"? Is it "ladies'" with the apostrophe after the s?

I ask this as a result of noticing the following bathroom signs at the Farm Show on Saturday:

<--- Ladies Men's --->

It caught my eye, but I wasn't sure what to say was wrong with it. I don't think it's wrong to have a bathroom for females labeled "Ladies," as the plural of "Lady," and I also don't think it's incorrect to label a bathroom for men "Men's," because the bathroom is for the Men so the posessive makes sense. But together, I think it's inconsistent to label one as plural only, and the other as plural posessive...UNLESS "Ladies" is already posessive? But I don't think it is...

What do you think? I think it would be best to just stick with plural and not posessive for both, "Ladies" and "Men"? But that does sound kind of weird because of the "s" on one. I guess I probably prefer "Men" and "Women" or else "Ladies" and "Gentelmen," just to keep a more traditional pair of words.

What a silly post this is...mainly seeing the two signs just got me wondering what is the plural posessive form of "Lady." To be clear, I'm not really bothered by this - my interest is purely academic!

In other news, the farm show was a lot of fun, very delicious, and even fairly guilt-free since Paul and I opted to walk the 1.2 miles uphill in the cold, snowy weather back to our car rather than waiting in the long long long lines for the busses to take us. Only disappointment for Paul was not being asked to judge the Pineapple Upside Down Cake competition!