  The bivariate chromatic polynomial $\chi_G(x,y)$ of a graph $G = (V, E)$,
introduced by Dohmen-P\"{o}nitz-Tittmann (2003), counts all $x$-colorings of
$G$ such that adjacent vertices get different colors if they are $\le y$. We
extend this notion to mixed graphs, which have both directed and undirected
edges. Our main result is a decomposition formula which expresses $\chi_G(x,y)$
as a sum of bivariate order polynomials (Beck-Farahmand-Karunaratne-Zuniga Ruiz
2020), and a combinatorial reciprocity theorem for $\chi_G(x,y)$.
