I ran into this question while trying to justify a claim made in Donald Monk's set theory notes that there exist countable transitive models of ZFC. A step in my sketch of this claim requires proving that if $(M,R)\vDash \phi$ for every $\phi\in\textrm{ZFC}$ then the relation $R$ on $M$ must itself be well-founded. I'm not sure if this actually holds or not.
Initially I thought that the well-foundedness axiom interpreted in $M$ may directly give the statement of well-foundedness for $(M,R)$ but this didn't work as well as I hoped. The well-foundedness axiom is given by $$ \textrm{WF}=\forall x (x \neq \varnothing \rightarrow \exists y\in x(y\cap x)=\varnothing). $$ Then $(M,R)\vDash \textrm{WF}$ is equivalent to $$ \forall x\in M (\exists z\in M (zR x) \rightarrow \exists y\in M (yRx \wedge \forall z\in M (zR y \rightarrow \neg zRx))). $$ The statement of well-foundedness for $R$ on the other hand is given by $$ \forall x\subseteq M (\exists y\in x \rightarrow (\exists w\in x \ \forall z\in x \ \neg zRw))). $$ Is there any way to establish well-foundedness from $(M,R)\vDash \textrm{ZFC}$? If not, will it this follow if we add the hypothesis that $M$ is transitive?