It is not so easy to show that the tropical multiplicity is well-defined, i.e. independent of the choice of $q$. Chambert-Loir and Ducros do not use tropical multiplicities, but the latter are equivalent to the canonical calibration introduced in [CD12], §3.5. To summarize this construction, let $e_{1},\dots,e_{n}$ (resp. $f_{1},\dots,f_{n}$) be a basis of $M^{\prime}$ (resp. $M_{\sigma}$). Then the canonical calibration of $\sigma$ is defined as

$$[\varphi_{\rm trop}^{-1}(\tau):{\rm Trop}(q)(\tau)]\cdot({\rm Trop}(q)|_{(N_{\sigma})_{\mathbb{R}}})^{*}(e_{1}\wedge\dots\wedge e_{n})\in\Lambda^{n}((N_{\sigma})_{\mathbb{R}})$$

together with the orientation induced by the pull-back of $e_{1},\dots,e_{n}$ with respect to the linear isomorphism ${\rm Trop}(q)|_{(N_{\sigma})_{\mathbb{R}}}$. The canonical calibration is equal to the calibration $m_{\sigma}f_{1}\wedge\dots\wedge f_{n}$ together with the orientation induced by $f_{1},\dots,f_{n}$. Since the canonical calibration does not depend on the choice of $q$ up to refinement ([CD12], §3.5), the same is true for the tropical multiplicities.