  The clustered chromatic number of a graph class $\mathcal{G}$ is the minimum
integer $c$ such that every graph $G\in\mathcal{G}$ has a $c$-colouring where
each monochromatic component in $G$ has bounded size. We study the clustered
chromatic number of graph classes $\mathcal{G}_H^{\text{odd}}$ defined by
excluding a graph $H$ as an odd-minor. How does the structure of $H$ relate to
the clustered chromatic number of $\mathcal{G}_H^{\text{odd}}$? Using a proof
method by Norin, Scott, Seymour and Wood (2019), we show that the clustered
chromatic number of $\mathcal{G}_H^{\text{odd}}$ is tied to the tree-depth of
$H$.
