Let $L$ be a countable language. Let $T$ be a complete $L$ theory. We know that if $T$ is small, then there is a prime model of the theory. But $\text{Th}(\mathbb{N},+,\times,0,1)$ is not small but it has a prime model.
As far as I'm aware there is no necessary and sufficient condition for the existence of prime models. But I would like to see how to show certain theories do not have prime models. For example, how would you show that any consistent extension of ZFC doesn't have a prime model? (I strongly suspect that is the case).
Edit 1: To clarify: I'm asking if there is a "standard" argument that you can try when you suspect a theory doesn't have a prime model.