Depends on which country. In places like India that’s not possible. Your cell phone number becomes a de facto identity so they require all kinds of identity documents to get a SIM.
So there's a cottage industry of middle men on the streets who will set you up with a SIM card, or a travel ticket or whatever, for people who don't have identity. (Or in some cases don't want to reveal their identity, but I reckon this is less typical.) Sure, you pay extra for the service, the middle man takes 10%, 30% or 500% and the identity is then with that person---or their fraudulent papers, I don't know how it works in detail.
It's usually the other way around - first countries introduce laws that require ID to buy a cell phone ("because criminals"), and then the phone number starts getting used as a de facto identity.
