  We say that a map $T: S_X\rightarrow S_Y$ between the unit spheres of two
real normed-spaces $X$ and $Y$ is a phase-isometry if it satisfies
\begin{eqnarray*} \{\|T(x)+T(y)\|, \|T(x)-T(y)\|\}=\{\|x+y\|, \|x-y\|\}
\end{eqnarray*} for all $x,y\in S_X$. In the present paper, we show that there
is a phase function $\varepsilon:S_X\rightarrow \{-1,1\}$ such that
$\varepsilon \cdot T$ is an isometry which can be extended a linear isometry on
the whole space $X$ whenever $T$ is surjective, $X=C(K)$ ($K$ is a compact
Hausdorff space) and $Y$ is an arbitrary Banach space. Additionally, if $T$ is
a phase-isometry between the unit spheres of $C(K)$ and $C(\Omega)$, where $K$
and $\Omega$ are compact Hausdorff spaces, we prove that there is a
homeomorphism $\varphi: \Omega\rightarrow K$ such that $T(f)\in\{f\circ
\varphi,-f\circ \varphi\}$ for all $f\in S_{C(K)}$. This also can be seen as a
Banach-Stone type representation for phase-isometries in $C(K)$ spaces.
