  For a locally compact Hausdorff space $L$, we denote by $C_0(L,\mathbb{R})$
the Banach space of all continuous real-valued functions on $L$ vanishing at
infinity equipped with the supremum norm. We prove that every surjective
phase-isometry $T\colon C_0^+(X,\mathbb{R}) \to C_0^+(Y,\mathbb{R})$ between
the positive cones of $C_0(X,\mathbb{R})$ and $C_0(Y,\mathbb{R})$ is a
composition operator induced by a homeomorphism between $X$ and $Y$.
Furthermore, we show that any surjective phase-isometry $T\colon
C_0^+(X,\mathbb{R}) \to C_0^+(Y,\mathbb{R})$ extends to a surjective linear
isometry from $C_0(X,\mathbb{R})$ onto $C_0(Y,\mathbb{R})$.
