  Given a graph $G$ without isolated vertices, a total Roman dominating
function for $G$ is a function $f : V(G)\rightarrow \{0,1,2\}$ such that every
vertex with label 0 is adjacent to a vertex with label 2, and the set of
vertices with positive labels induces a graph of minimum degree at least one.
The total Roman domination number $\gamma_{tR}(G)$ of $G$ is the smallest
possible value of $\sum_{v\in V(G)}f(v)$ among all total Roman dominating
functions $f$. The total Roman domination number of the direct product $G\times
H$ of the graphs $G$ and $H$ is studied in this work. Specifically, several
relationships, in the shape of upper and lower bounds, between
$\gamma_{tR}(G\times H)$ and some classical domination parameters for the
factors are given. Characterizations of the direct product graphs $G\times H$
achieving small values ($\le 7$) for $\gamma_{tR}(G\times H)$ are presented,
and exact values for $\gamma_{tR}(G\times H)$ are deduced, while considering
various specific direct product classes.
