In this work, we study the $δ$-chromatic number of a graph which is the chromatic number of the $δ$-complement of a graph. We give a structure of the $δ$-complements and sharp bounds on the $δ$-chromatic numbers of the Cartesian products of graphs. Furthermore, we compute the $δ$-chromatic numbers of various classes of Cartesian product graphs, including the Cartesian products between cycles, paths, and stars.