An acquaintance recalled this specific example from an English textbook, but it is jarring to my native ear. Is this an example of prescriptive grammarians gone wild?
It is grammatical, but it is indeed extremely jarring. It is (to me at least) just as jarring (if not more so) to say
A much better way to express the idea is to say
On what basis do I say that it is grammatical, if it is so jarring?
Since me is undoubtedly the first person singular, the verb to go with it has to be am. So yes, jarring though it is, the phrase is correct.
I think the unusual part of the phrase is that me, though the object of the sentence, is the subject of the subordinate clause, and the reader naturally tries to read to be as the main verb.
The underlying problem with this sort of construction is that we are essentially condensing two sentences into one, and forcing a dual subjective and objective role on the linking word in the ellipted form.
I am your friend.
Remember me, who am your friend. – here, me is the object of the (recommended) remembering, and the at least notional (some would claim not grammatically speaking) subject of the copula.
Unless a way of avoiding or at least hiding the conflict is found (eg reducing to an apposition, as psmears wisely advises), a rule to essentially allow one existing rule to be broken has to be fudged. Who will probably be claimed as the true subject of the copular construction.)
Another example in this area is the famous
The prescriptive 'over-rule' is that he/him (who is without sin), which links the letting and the casting the first stone, has its case governed by the letting (ie is in the objective) rather than by the casting the first stone. But (though hits are surprisingly low) Google stats indicate the 'incorrect' choice occurs three times as often as the 'correct'. Will the rule change!?
(In this case, the problem can be neatly hidden by substituting the uninflecting 'the man' for him/*he.)
Another rule is claimed for the relative pronouns themselves:
[This gem appears at Who/whom | Facebook , thankfully followed by:]
And with whoever / whomever:
Let whomever is without sin cast the first stone.
Let whoever is without sin cast the first stone.
Let whomever that stone hit sue the perpetrator.
Fowler’s Dictionary of Modern (!?) English Usage, 1994 ed, agrees with the revisers who produced the RV of the Bible that the AV's:
“Eye hath not seen, nor ear heard, neither have entered into the heart of man, the things which God hath prepared for them that love him.”
demands too much grammatically of “the things” in requiring it to be both object of the two verbs seen and heard as well as the subject of entered.
I don't think this is grammatical. Not only have I never seen an example of this construction in the wild (of course, I don't go looking for such things), I don't think it really parses sensibly.
'Am' goes only with the first person singular subject, so for this to be correct, it would have to be possible to replace "Remember me, who" with "I". Given that the construction already has "me", that phrase is more naturally interpreted as the object, which takes "is" in agreement with the third person.