I am not sure that this question even makes sense, which I suppose is part of the questions itself.

In any case, I attended a talk recently wherin there was some discussion about a "tropical Teichmuller space", as for example in this paper, associated to the tropical moduli space of abelian varieties. My understanding is that the moduli space of tropical abelian varieties $A_g^{trop}$ can be identified with the skeleton of a Berkovich analytic space associated to the moduli space of abelian varieties $A_g$. Does the Hodge Bundle on the moduli space of abelian varieties "carry over" (whatever that should mean) to the analytic space? If so, can you then "pull it back" to the skeleton, onto which the analytic space deformation retracts?

My question is motivated by the fact that the speaker suggested that this "tropical Teichmuller space" gives some tropical analogy to the Siegel upper half space model for $A_g$, where you take a quotient by $GL_n$ to get the moduli space of tropical abelian varieties. I am basically wondering if there is some notion of a "tropical Siegel modular form", which arises as a section of a "tropical Hodge bundle".

Some nice references for tropical geometry and Berkovich analytic spaces is also welcome.

Note: I also posted this question here on StackExchange, but received no answers.