Let $S$ be a finite set, let $d \in \mathbb{N}$, and let $X \subset S^{\mathbb{Z}^d}$ be a compact shift-invariant configuration space with the [product topology](/page/Product%20Topology) and the $\mathbb{Z}^d$-shift action. Let $\beta: X \to \mathbb{R}$ be a continuous potential, and let $\Phi$ be a finite-range interaction on $X$. Assume that, for every shift-invariant Borel probability measure $\mu$ on $X$, the measure $\mu$ is an equilibrium state for $\beta$ if and only if $\mu$ is a shift-invariant DLR Gibbs state for $\Phi$. If $\beta$ has at least two distinct shift-invariant equilibrium states, then $\Phi$ has a phase transition; equivalently, the shift-invariant DLR Gibbs states for $\Phi$ are not unique.