An integral Kähler affine structure on $Y$ is bi-polyhedral (bi-PIKAS for short) if there is a bipartite covering $\{U_{\alpha},V_{\beta}\}$ of $Y$ and two polyhedral complexes $P,\hat{P}$ such that $(\{U_{\alpha}\},P)$ and $(\{V_{\beta}\},\hat{P})$ provide polyhedral realizations of the underlying affine structure and its dual, respectively. We say that the bi-polyhedral Kähler affine structure is of type $(\{U_{\alpha},V_{\beta}\},P,\hat{P})$.