They handle it just fine, of course. I disagree with the statement that Japanese has a relatively high incidence of homophones. It is true that Japanese TV shows, particularly variety shoes and lighter fare like that, have a lot of on-screen captions and I don’t know the reason for that, but I doubt it’s homophones, and at any rates more “serious” shows like dramas as well as all movies are caption-free (but will likely have a subtle track you can enable).