3Col reduction Variation, Special edges
You can reduce from $3$-coloring.
Given an instance (graph) $G=(V,E)$ of $3$-coloring, create a tri-graph $G'=(V', E')$ by transforming each edge $e=(u,v) \in E$ to a tri-edge $(u,v,z_e)$ in $G'$ (where $z_e$ is a new vertex).
A 3-coloring $c : V' \to \{1,2,3\}$ of $G'$ induces a $3$-coloring of $G$: simply color $v \in V$ with color $c(v)$.
A 3-coloring $c : V \to \{1,2,3\}$ of $G$ can also be transformed into a $3$-coloring of $G'$ by coloring each vertex $z_e$, where $e=(u,v) \in E$, with the unique color in $\{1,2,3\} \setminus \{c(u), c(v)\}$.