Well ... the nearest thing I know of are the reflexive pronouns in Danish. Specifically the possesives.

(This is a 'dying' trait, though.)

Han tog hans hat og gik - He took his hat and left.

Han tog sin hat og gik - He took his-own hat and left.

So not really the same thing, but I don't know many languages.

Stop me if I'm wrong, but don't Georgian verbs agree with their object as well as their subject?

Conceptually similar, but not as obtruse, is the mistake you occasionally hear from Romance-speakers in German; when using possessive pronouns, they will use the gender of the possessed thing, but ignore the gender of the possessor. Whereas we English-speakers make the opposite mistake, ignoring the gender of the possessed thing, but not the gender of the possessor. For example, said of woman:

»So war seine Geschichte« — Romance-speaker mistake
»So war ihr Geschichte« – English-speaker mistake.
»So war ihre Geschichte« – Correct version.

My faulty memory suggests that in one of Borges' fictions, a language was discussed which had different grammatical forms depending on the status relationship between the speaker and the spoken-to.

"a language was discussed which had different grammatical forms depending on the status relationship between the speaker and the spoken-to."

Japanese and Korean both do that. Some ethnic languages of Indonesia do that lexically, so the Sundanese word for 'eye' is panon if you're speaking to a friend and soca if you're speaking to a senior.
No every word has the equivalants but it's easy to get more than one word in a sentence that does:
Urang indit ka Bandung.
Abdi angkat ka Bandung.
Both mean "I'm going to Bandung" but the first is familar and the second more politce (again, speaking to those more important than you).

If I understand correctly the most interesting example is:

đālikumā mimmā `allamanī rabbī
"That is (part) of what my Lord has taught me."

because the 'you' marked with -kumā is not otherwise a sentence constituent ...

And no, I can't think of any other language that does that.

don't Georgian verbs agree with their object as well as their subject?

Yes, and that happens in other languages as well, but this question is specifically about relative pronouns.

Huh -- so perhaps I was reading about Japanese and Korean, or about Indonesian languages, and when I happened on this fact I filed it away mentally as "trippy Borgesian idea"... at some point the night staff of my memory saw that file and decided it might as well be mixed in with the "Borges' fictions" drawer.

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I was reading one of those fluffy intercultural communication books for world travellers and expats, and it noted that Indonesians are more or less the only foreigners who ever master Japanese politeness forms.

Basque does things like that. I'm not far enough advanced to understand how it works, but i understand there are inflections based on the addressee.

AFAIK Basque "just" has verb agreement with subjects and objects.

Well ... the nearest thing I know of are the reflexive pronouns in Danish. Specifically the possesives.

Judging from your examples, it looks like one is the genitive of the personal pronoun, and the other is the possessive pronoun. Russian also uses both, and so did Latin (его, eius; свой, suus). In German, the genitive of the personal pronoun is only used with certain prepositions, and even that only in elevated style that borders on the archaizing.

Thanks for posting this! It's brought up a few interesting examples of multiple agreement phenomena, but so far my impression that demonstratives agreeing with the addressee are rather rare seems to be borne out. This may have some implications for the history of Siwi, but I'm still working on that...

Basque was brought up in the first comment to the original post. See there and this page's Table 2:

A more unusual property of this gender-marking is its allocutivity, by which the inflected verb agrees with addressee's gender, even when addressee is not in sentence.

In Jabêm, a well-documented language of Papua New Guinea, relative clauses are usually marked at both ends by demonstrative formants. The demonstratives correlate with 1st, 2nd, or 3rd person, and the choice of which set to use to mark a particular clause indicates whether the reference is assumed to be known to either the speaker, the hearer, or neither. The same is true in other area languages, but as a practical matter more people rely on less indicative clause markers, including the equivalent of WH-forms in English ('where, which').

"Hans (his)" is indeed the genitive of "han (he)" - correspondingly "hendes (her)", "hun (she)" in the feminine.

The point was that those are only used to refer to the possessions of a third party - if referring back to the subject of the sentence you'd use "sin" (or "sit" depending on gender). A distinction not generally made in English (I believe) and on its way out in Danish.

A number of people have mentioned Basque. Larry Trask's book on Basque mentions that some dialects of Basque have verb markers which agree with the (gender?) of the addressee even when the addressee is not one of the arguments of the verb. This is distinct from, and in addition to, 2nd person singular markers which Basque verbs also have when the 2nd person is one of the arguments of the verb. Trask says that these markers are only used between people have known each other all their lives e.g. old school friends, but I'm not sure it includes parents. I'm quoting from memory because I don't have Trask's book.

The point was that those are only used to refer to the possessions of a third party - if referring back to the subject of the sentence you'd use "sin" (or "sit" depending on gender). A distinction not generally made in English (I believe) and on its way out in Danish.

So does han tog hans hat og gik mean "he took the other guy's hat and left"? That distinction is commonly (though not always) made in my dialect, and it could be interesting that it's done in one of the ways to replace the missing genitive.

Exactly!