For every maximal dual $\Delta$-complex $\Pi$ there exists a smooth hypersurface $V\subset\mathbb{C}T_{\Delta}$ defined over $\mathbb{R}$ such that the map $\lambda$ from Theorem 1’ preserves the real structure of $V$, i.e. $\lambda\circ\operatorname{conj}=\lambda$. Furthermore, $\lambda$ is a totally real fibration.