Positivity of tropical multidegrees

We define the multidegrees of a tropical variety. We prove that the positivity of a multidegree of a certain tropical variety is governed by the dimensions of the images of the tropical variety under suitable projection maps. As an application, we give a tropical proof of the criterion of the positivity of the multidegrees of a closed subscheme of a multi-projective space, originally proved by Castillo et al.