  Given a graph $G$, a mutual-visibility coloring of $G$ is introduced as
follows. We color two vertices $x,y\in V(G)$ with a same color, if there is a
shortest $x,y$-path whose internal vertices have different colors than $x,y$.
The smallest number of colors needed in a mutual-visibility coloring of $G$ is
the mutual-visibility chromatic number of $G$, which is denoted
$\chi_{\mu}(G)$. Relationships between $\chi_{\mu}(G)$ and its two parent ones,
the chromatic number and the mutual-visibility number, are presented. Graphs of
diameter two are considered, and in particular the asymptotic growth of the
mutual-visibility number of the Cartesian product of complete graphs is
determined. A greedy algorithm that finds a mutual-visibility coloring is
designed and several possible scenarios on its efficiency are discussed.
Several bounds are given in terms of other graph parameters such as the
diameter, the order, the maximum degree, the degree of regularity of regular
graphs, and/or the mutual-visibility number. For the corona products it is
proved that the value of its mutual-visibility chromatic number depends on that
of the first factor of the product. Graphs $G$ for which $\chi_{\mu}(G)=2$ are
also considered.
