The correspondence $\Pi\mapsto\operatorname{c}(\Pi)$ is a bijection between the set of complete polyhedral complexes in $N_{\mathbb{R}}$ and the set of complete conical polyhedral complexes in $N_{\mathbb{R}}\times\mathbb{R}_{\geq 0}$. Its inverse is the correspondence that, to each conic polyhedral complex $\Sigma$ in $N_{\mathbb{R}}\times\mathbb{R}_{\geq 0}$ corresponds the complex in $N_{\mathbb{R}}$ obtained by intersecting $\Sigma$ with the hyperplane $N_{\mathbb{R}}\times\{1\}$. These bijections preserve rationality and strong convexity.