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If $G$ is a graph and $X\subseteq V(G)$, then $X$ is a totalmutual-visibility set if every pair of vertices $x$ and $y$ of $G$ admits ashortest $x,y$-path $P$ with $V(P) \cap X \subseteq \{x,y\}$. The cardinalityof a largest total mutual-visibility set of $G$ is the total mutual-visibilitynumber $\mu_{\rm t}(G)$ of $G$. Graphs with $\mu_{\rm t}(G) = 0$ arecharacterized as the graphs in which no vertex is the central vertex of aconvex $P_3$. The total mutual-visibility number of Cartesian products isbounded and several exact results proved. For instance, $\mu_{\rmt}(K_n\,\square\, K_m) = \max\{n,m\}$ 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 arbitrarylarger than $\mu_{\rm t}(G)\mu_{\rm t}(H)$.

Graphs with total mutual-visibility number zero and total mutual-visibility in Cartesian products