Painted Tropical Complexes

We define the notion of a painted tropical $A$-complex and describe a poset structure on the set of all such complexes. This poset is equivalent to the face lattice of a secondary polytope $\Sigma (\bar{A}_\alpha )$ where $\bar{A}_\alpha$ is built from $A$ and an additional point $\alpha$. As a central application, we show that multiplihedra are also secondary polytopes.