  If $G$ is a graph and $X\subseteq V(G)$, then $X$ is a total
mutual-visibility set if every pair of vertices $x$ and $y$ of $G$ admits a
shortest $x,y$-path $P$ with $V(P) \cap X \subseteq \{x,y\}$. The cardinality
of a largest total mutual-visibility set of $G$ is the total mutual-visibility
number $\mu_{\rm t}(G)$ of $G$. Graphs with $\mu_{\rm t}(G) = 0$ are
characterized as the graphs in which no vertex is the central vertex of a
convex $P_3$. The total mutual-visibility number of Cartesian products is
bounded and several exact results proved. For instance, $\mu_{\rm
t}(K_n\,\square\, K_m) = \max\{n,m\}$ and $\mu_{\rm t}(T\,\square\, H) =
\mu_{\rm t}(T)\mu_{\rm t}(H)$, where $T$ is a tree and $H$ an arbitrary graph.
It is also demonstrated that $\mu_{\rm t}(G\,\square\, H)$ can be arbitrary
larger than $\mu_{\rm t}(G)\mu_{\rm t}(H)$.
