  We give a purely tropical analogue of Donagi's $n$-gonal construction and
investigate its combinatorial properties. The input of the construction is a
harmonic double cover of an $n$-gonal tropical curve. For $n = 2$ and a dilated
double cover, the output is a tower of the same type, and we show that the Prym
varieties of the two double covers are dual tropical abelian varieties. For
$n=3$ and a free double cover, the output is a tetragonal tropical curve with
dilation profile nowhere $(2,2)$ or $(4)$, and we show that the construction
can be reversed. Furthermore, the Prym variety of the double cover and the
Jacobian of the tetragonal curve are isomorphic as principally polarized
tropical abelian varieties. Our main tool is tropical homology theory, and our
proofs closely follow the algebraic versions.

Combinatorics;Algebraic Geometry