In the following example, what tense of the verb should be used: past or non-past? If both are possible will the meaning change?
Children will eat anything their mother brings home.
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In both sentences, the main verb is 「食べます」 and the tense of the main verb is the tense of the sentence.
That means whether you use 「母親が持ち帰る」 or 「母親が持ち帰った」 as a relative clause to modify the 「もの」, it has no effect at all on the tense of the sentence itself, which is present. In other words, it is not of much importance because it does not change the meaning of the sentence.
It is nothing like the choice between 「食べます」 and 「食べました」, which will change the meaning of the sentence in a major way.
I just could not find a difference worth mentioning between 「持ち帰る」 and 「持ち帰った」. You would probably hear 「持ち帰った」 slightly more often from us native speakers. If anything, using 「持ち帰った」 would make the sentence sound just a wee bit more informal or colloquial, but again, the difference is minimal and people would not even notice it most of the time.