Bounded Twin-Width Tournaments are $\text{dchi}$-Bounded

For a tournament $T$ and a vertex ordering $\prec$, let $T^{\prec }$ be the graph of backward arcs in $\prec$. The diclique number of a tournament is $\text{domega}(T)={\min }_{\prec }\omega (T^{\prec })$, and the dichromatic number is $\text{dchi}(T)={\min }_{\prec }\chi (T^{\prec })$. We prove a mixed parameter transfer theorem: for all $k$ and $r$, if $\text{tww}(T)\le k$ and $\omega (T^{\prec })\le r$, then the ordered twin-width of $(T^{\prec },\prec )$ is bounded by a function of $k$ and $r$. The proof combines the regular-semigrid theorem for ordered graphs with permutation-encoding obstructions to bounded twin-width in tournaments. Together with polynomial $\chi$-boundedness of graphs of bounded twin-width, this implies that tournaments of bounded twin-width are $\text{dchi}$-bounded by $\text{domega}$, resolving a conjecture of Aboulker, Aubian, Charbit, and Lopes.