I usually encounter this with pest cold-calls, preceded by a robotic greeting and query. Somebody comes on with a Hindi accent that could run from Mumbai to Calcutta. I just call them out on having an impenetrable accent and that I have no interest in talking to someone I can't understand. I think that offends them, but I have rung off by that point.

Another modern symptom is their habit of opening with "Hello, (my first name)." I usually take this as an opportunity to remind them that in English-speaking countries, it is proper manners of respect to address a stranger as "Mr. (surname)," as we are not on friendly acquaintance. Then I say, "Which (first name) do you wish to speak with?" They are usually servile enough to comply with the education.

But, honestly, there are plenty of native English-speakers that have terrible enunciation and do not speak up. Most annoying are the young women who speak in cozy, murmuring voices, trying to be demure or something. It is more like listening to winds on a distant seashore.