Which is correct grammatically - "women's greatest asset is their body" or "women's greatest asset are their bodies"?

None of those are correct. The correct phrasing is "Women are inferior to Men because inherently the only worth they have is their body" 

Women's greatest assets are their bodies.

Neither... It's 'women's only asset are their bodies'... No body and they are useless most of the time.. 

You can't say "AN asset ARE."  

Correct:  a woman's greatest asset is her body.  And that's a crock.  A women's greatest asset is her MIND.

Both options are wrong. You started your sentence with the word ‘women’s’ so your sentence has to say, women’s greatest assets are their bodies. Because you said women, there is more than one asset. 

Both are incorrect.  " A woman's greatest asset is HER  body" is correct. Also not true) 

Although all of the woman could have exactly the same asset, they cannot have the same body.  therefore, the asset is not a unique asset and should be plural if you are discussing multiple women.  Women's greatest assets are their bodies.  The sentence does not impose sharing of bodies among different women as that is impossible.

The only way you can use a singular asset in that sentence (or any similar sentence) is if the asset is a singular item that each and all of the "women" (in that sentence) are owners of.  there could only by ONE asset.  Well, there is more than one body, so there is not one asset.

If three women owned a house, for example, then you could say "the women's greatest asset was the house".  There is only one house, one asset.  If each woman had her own house, then the women's greatest assets would be their own houses.

imo the more common phrase is the singular  "A woman's greatest asset is her body." That way you are but are not labeling all women (as worthless). [<yeah, mixed message !]

Let's ask, "Whose greatest asset ?".The answer coming out is " women's assent".Here " asset" is the subject, a singular number.For this, the verb ought to be singular, that is "is" in place of "are".." asset" is compared to "women's bodies". Above all the final structure will be......

   Women's greatest asset is their bodies.  You too referred this in your updated phrase.

Thanks.

"Women's greatest assets are their bodies."

OR: "A woman's greatest asset is her body."  Possibly this form might be more usually seen in Britain.  Await answers from Americans etc.

Plural must follow plural, or singular must follow singular.

But don't shout the idea too loudly in these days of women's lib!

The first is correct and the second nedds a change to "is their bodies" (the subect of the verb 'are' is 'asset', which is singular, so 'are' should be changed to 'is').