Crossing tournaments are polynomially $\vec{\chi }$-bounded

Given a tournament $T$, Aboulker, Aubian, Charbit, and Lopes (2023) defined its clique number $\vec{\omega }(T)$ as the minimum clique number of a backedge graph of $T$, and raised the question: Which classes of tournaments are polynomially $\vec{\chi }$-bounded? Aboulker, Duron, Jacob, Kimbrough, Thomassé, and this work’s authors (2026) showed that this holds for classes of tournaments whose arc sets may be written as the union of a bounded number of comparability digraphs. What about classes of tournaments that do not admit such a decomposition? The crossing tournaments of Nguyen, Scott, and Seymour (2025) are an example of such a class, as shown in the aforementioned 2026 work; we show that nonetheless crossing tournaments are polynomially $\vec{\chi }$-bounded by adapting a method of Davies and McCarty (2021) and Davies (2022). We additionally show that we cannot extend this result for crossing tournaments to tournaments with chordal graphs as backedge graphs.