The NZ Folksong page on "Pokarekare Ana" reports a suggestion that the tune might have been of Croatian origin. It certainly sounds a lot like a Dalmatian "klapa" choral song, and the north end of the North Island was settled by migrants from Dalmatia in the generation before the Maori song was composed, so transmission is historically plausible.

But I've asked my Croatian contacts if they recognize it and got nowhere. Anybody know folks in a klapa group to ask?