One example of such awkwardness and futility is The Roman Spring of Mrs. Stone (1961): if Warren Beatty's unbearable fake Italian accent is to present some sort of realism/realisticity, the "realism/realisticity" is undone when the Italian characters talk to each other in English (with fake Italian accent).

There's another kind of futility: for example, when an Anglophone film is set in Ancient Rome or Greece, what difference does British accent make from American accent in terms of Greco-Roman authenticity?

Don't get me wrong - I'm fine with any vernacular film set in Ancient Rome or Greece, or wherever: I'm fine with Greco-Roman characters speaking perfect English in an Anglophone film, I'm fine with Anglophone characters speaking perfect French in a Francophone film, I'm fine with Francophone characters speaking perfect Russian in a Russophone film ... those who have a problem with that, imho, are hypercritical.