  Let $G$ be a graph and $X\subseteq V(G)$. Then $X$ is a mutual-visibility set
if each pair of vertices from $X$ is connected by a geodesic with no internal
vertex in $X$. The mutual-visibility number $\mu(G)$ of $G$ is the cardinality
of a largest mutual-visibility set. In this paper, the mutual-visibility number
of strong product graphs is investigated. As a tool for this, total
mutual-visibility sets are introduced. Along the way, basic properties of such
sets are presented. The (total) mutual-visibility number of strong products is
bounded from below in two ways, and determined exactly for strong grids of
arbitrary dimension. Strong prisms are studied separately and a couple of tight
bounds for their mutual-visibility number are given.
