  In this paper, we present a unified study of the moduli space of tropical
curves and Outer space which we link via period maps to the moduli space of
tropical abelian varieties and the space of positive definite quadratic forms.
Our work is a first step towards exhibiting Outer space and the space of
positive definite quadratic forms as analogues of Teichm\"uller space and
Siegel space, respectively, in tropical geometry. All these spaces and the maps
among them are described within the category of ideal stacky fans, which we
describe in detail.
