"Each other" - singular or plural in this sentence

In this sentence, should the last word be "life" or " lives"?

(Imagine two children brought up by the wrong parents to make this sentence make sense!)

They had had each other's life/lives.

Many thanks for a definitive answer!

According to the corpora 1. English Web 2018: https://app.sketchengine.eu/ 2. UKWac Complete: https://corpora.dipintra.it/public/run.cgi/first_form 3. Google Books Ngram Viewer: https://books.google.com/ngrams , "each other's/one another's lives/faces/throats/hands" is correct while "each other's/one another's life/face/throat/hand" is incorrect.

Gngram has hits for both, so it looks like both are correct, but the plural is more common, probably because people find it more reasonable since each other involves at least two persons or things (some would argue that only two persons are involved), therefore, whatever these two possess should be in the plural.

In my view, you speak of something that each person does to or has of the other. So life will perfectly make sense here, it is in fact the most logical of the two versions. If I logically reformulate the sentence to mean the same thing, here's what I get:

They each had had the other's life.

OR

They both ended up having a different life, the life of the other.

I don't think anyone would misunderstand. We are talking about two lives, but each one had one life, not two.

There is an interesting argument for this use on this site. I know it is not official, yet it does shed some light on the matter in a logical way of thinking.

But, ok, it's not a "reliable source", so let's find one. The Cambridge Dictionary gives another example that proves that you can put either plural or singular after each other's (or one another's - the dictionary says the two are interchangeable), depending maybe on what you want to stress, or just because it sounds more natural to you.

• My friend Olivia and I know very little about one another’s work. (Cambridge)

Edit: Prompted by one comment I am giving a better source for the use with singular, countable nouns:

We are one team and won't be thinking too much about taking each other's place. Times, Sunday Times (2016) (Collins)

The two families, while watching each other's back, can also prove instrumental in each other's destruction. (The Guardian)

Unique agreement includes special Young Professionals scheme to allow young Indian and British professionals to work and live in each other’s country for two years... (gov.uk)

I found another very interesting explanation of this matter on another grammar site which offers reliable sources:

The traditionalists are at odds here. In The Careful Writer Theodore M. Bernstein claims that each other’s is equivalent to their. So Bernstein would say The lawyer and the banker admired each other’s cars. But Bryan A. Garner leans toward the singular car. In A Dictionary of Modern American Usage, Garner says “the noun that follows is often plural <each other’s cars>, but the more logical construction is singular <each other’s car>.”

I couldn't help smiling when reading the next line:

Did he say “logical”? When it comes to each other and one another, logic is beside the point.

So let's not argue about this, it's "beside the point"!

The "logical" conclusion? You can use either, as there are reasonable arguments for both variants, the prevailing one being probably that of use. People say much more often each other's lives, and that's fine. Just don't dismiss the other variant as incorrect and illogical, cause it's not.