Let $\Pi$ be the polyhedral complex in $\mathbb{R}^{3}$ containing the faces of the polyhedra

$$\Lambda_{1}=\{(x_{1},x_{2},0)|\,x_{1},x_{2}\geq 0\},\quad\Lambda_{2}=\{(x_{1},x_{2},1)|\,x_{1}+x_{2},x_{1}-x_{2}\geq 0\}.$$

Then $\operatorname{rec}(\Lambda_{1})$ and $\operatorname{rec}(\Lambda_{2})$ are two cones in $\mathbb{R}^{2}\times\{0\}$ whose intersection is the cone $\{(x_{1},x_{2},0)|x_{2},x_{1}-x_{2}\geq 0\}$. This cone is neither a face of $\operatorname{rec}(\Lambda_{1})$ nor of $\operatorname{rec}(\Lambda_{2})$. Hence $\operatorname{rec}(\Pi)$ is not a complex and, consequently, neither is $\operatorname{c}(\Pi)$. In Figure 1 we see the polyhedron $\Lambda_{1}$ in light grey, the polyhedron $\Lambda_{2}$ in darker grey and $\operatorname{rec}(\Lambda_{2})$ as dashed lines.