I used Settings > General > Reset > Reset All Settings on my iPhone, and now incoming phone calls are displayed as the telephone number, rather than the name from my Contacts.
I do notice that the telephone numbers in my Contacts app are (now?) simply a string of 10 digits, rather than being formatted. I'm pretty sure it was formatted at one point.
Thought it might have something to do with International Assist, but turning that on or off doesn't change this.
I don't recall having to set anything initially to enable caller i.d., so I don't know what setting might have been reset.