Is "each's" a word?

According to Fowler's Dictionary of Modern English Usage (2015; link), each's is

an uncommon but acceptable use: The distressing conflict between Catholics and Jews… is driven by each's belief that the other is attempting to [etc.] Chicago Tribune, 1989.

Furthermore, each's is listed together with another's, each other's, either's, one's, one another's, other's, etc. in A Concise Guide for Writers by Louis E. Glorfeld, David A. Lauerman, Norman C. Stageberg, 1984 (link)

CGEL is a bit less certain. It says the following (p. 420, boldface emphasis mine)

The forms we analyse as fused heads either cannot occur in the genitive at all or do so only very rarely. ^*All's, ^*many's, ^*few's, ^*some's, for example, are completely excluded, while each's is occasionally attested but of somewhat questionable status. A pronoun analysis provides no account of why this should be so, while the fused analysis does suggest a reason: the form is simultaneously determiner and head, and for full acceptability its case should match both functions. The genitive marking in many people's expenses goes on the head, not the determiner, and hence there is a conflict between case and determiner function in ^*many's expenses.

Finally, if you search google books for "each's" (like this), you will find quite a few hits in publications written by native speakers.

Some examples:

Radial distance of the peak position, midpoint position and peak - to - midpoint difference from each's primary only average for each delay for Data Set 1
_{From Anthony R. Valenzuela, Relativistic Free Electrons in an Intense Laser Field: Experimental Observations of Optically-induced Deflection of an Ultrashort Electron Beam, (2005) (link)}

We feel that he is justified in fearing each's suffering, looking forward to each's pleasures, planning for each's future, and so on.
_{From Jeff McMahan, The Ethics of Killing: Problems at the Margins of Life, (2001) (link)}

….immediate and full reinstatement to each's former position or, if such position no longer exists, to a substantially equivalent position, without prejudice to each's seniority, or other rights previously enjoyed…
_{From Decisions and Orders of the National Labor Relations Board, Volume 256 (1981) (link)}

It is true that the verdict that each's is acceptable is not unanimous. For example, in The Practical Guide to Writing: with Readings and Handbook by Sylvan Barnet (2007, link), it is stated that

Each cannot be made into a possessive; you cannot say "Each's opinion is acceptable."

The other two 'big' modern comprehensive descriptive grammars of English (ComGEL and Longman) do not say anything one way or another about each's. Neither do Garner's Modern American Usage, The Chicago Manual of Style, or The Associated Press Stylebook.

On the whole, the balance of evidence suggests that each's is generally acceptable, albeit uncommon. Having said that, there will be native speakers who will find it nearly unacceptable.