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 $\mathrm{\Sigma }({\bar{A}}_{\alpha })$ 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.