Using the notation from above, the tropical multiplicity $m_{\sigma}$ along $\sigma$ is defined by

$$m_{\sigma}:=[\varphi_{\rm trop}^{-1}(\tau):{\rm Trop}(q)(\tau)]\cdot[M_{\sigma}:M^{\prime}]^{-1}.$$

Furthermore, $({\varphi_{\rm trop}})_{*}({\rm cyc}(Z))$ is the weighted polyhedral complex ${\mathscr{C}}$ endowed with these tropical multiplicities.