I've seen people putting the possessive adjective "min" both before and after the noun they refer to. I'm confused, is there a rule for this?

質問を翻訳

"min, mi, mitt, mine, din, di, ditt, dine, sin, si, sitt, sine" etc. are pronouns, not adjectives. You're right, however, in that they are posessive, and that they can appear either before or after the referent noun. Two different ways of forming the possessive exist:

1. [possessive pronoun] + [indefinite noun]
2. [definite noun] + [possessive pronoun]

A minor complication to this is that the former scheme is only compatible with the two-gender system, while the latter fits both two and three genders. If you use three genders generally, you can use #1, but you have to use the common gender for masculine and feminine nouns in this specific case. In practice the only difference is that "mi, di, si" merge into "min, din, sin".

Ostensibly it is up to the speaker/writer which of the two possessive systems they use at any given time. Dialect is the primary decider, but context is also important. For instance, #1 is somewhat more common when the possessive reveives particular emphasis.

In a lot cases one of the two will sound unnatural, but both are ostensibly correct in most cases; #1 is probably the safer bet in terms of being technically correct, but it will likely sound very stilted a lot of the time. A more effective approach might be to use #2 as general purpose, and substitute it with #1 in special cases, such as when receiving specific stress, using archaic or poetic language, or using fixed turns of phrase or terms of endearment.