A set $S$ of vertices in a graph $G$ is a dominating set if every vertex of $V(G) \setminus S$ is adjacent to a vertex in $S$. A coalition in $G$ consists of two disjoint sets of vertices $X$ and $Y$ of $G$, neither of which is a dominating set but whose union $X \cup Y$ is a dominating set of $G$. Such sets $X$ and $Y$ form a coalition in $G$. A coalition partition, abbreviated $c$-partition, in $G$ is a partition $\mathfrak{X} = \{X_1,\ldots,X_k\}$ of the vertex set $V(G)$ of $G$ such that for all $i \in [k]$, each set $X_i \in \mathfrak{X}$ satisfies one of the following two conditions: (1) $X_i$ is a dominating set of $G$ with a single vertex, or (2) $X_i$ forms a coalition with some other set $X_j \in \mathfrak{X}$. Given a coalition partition $\mathfrak{X}$ of a graph $G$, a coalition graph CG$(G,\mathfrak{X})$ is constructed by representing each member of $\mathfrak{X}$ as a vertex of the graph, and joining two vertices with an edge if and only if the corresponding sets form a coalition in $G$. If each set in a coalition partition $\mathfrak{X}$ of $G$ contains only one vertex, then $\mathfrak{X}$ is referred to as a singleton coalition partition. A graph $G$ is a complementary coalition graph if CG$(G,\mathfrak{X})$ is isomorphic to the complement of $G$. We characterize complementary coalition graphs. This solves an open problem posed by Haynes et al. [Commun. Comb. Optim. 8 (2023) 423–430]. Moreover, we provide a polynomial-time algorithm that determines if a given graph is a complementary coalition graph.