I recently saw someone intentionally use "eachother" instead of "each other". In what circumstances would this be correct?
In my view this is not a valid alternate for "each other" under any circumstances. I would go as far as saying it's plain incorrect.
Saying that, Wiktionary lists it as a "non-standard spelling", though that's an understatement. Neither the OED nor Webster's lists it at all. I have certainly never seen it used by any native speaker of English in my life.
I don't remember seeing this before. If I was given some English text to correct and saw "eachother" I would certainly change it to "each other".
But, looking at the Wiktionary there are quite a few examples:
Google books turns up a lot more hits, including the above, which seem to date back a fair way. An interesting one is this English grammar book from 1814.
There are 8,900,000 hits for "eachother" on Google compared to 135,000,000 for "each other".
So it seems to be a kind of minority/historical spelling.
I've never come across "eachother" before, and would dare to claim it isn't ever correct, but more like a spelling mistake.
Or, to put it more "descriptively" (as seems to be customary), "eachother" at least is not standard in any common variety of English. :-)
Some statistical evidence: The Corpus of Contemporary American English (COCA) finds 10 occurrences of "eachother" (versus 56,000 for "each other"). Also historically it has enjoyed only very sporadic use: Corpus of Historical American English (COHA) lists 24 (vs 60,000) occurrences in material from 1810 to 2009. I wouldn't say this qualifies "eachother" as correct.