  In 2006, Kenyon and Okounkov computed the moduli space of Harnack curves of
degree $d$ in $\mathbb{C}\mathbb{P}^2$. We generalize to any projective toric
surface some of the techniques used there. More precisely, we show that the
moduli space $\mathcal{H}_\Delta$ of Harnack curves with Newton polygon
$\Delta$ is diffeomorphic to $\mathbb{R}^{m-3}\times\mathbb{R}_{\geq0}^{n+g-m}$
where $\Delta$ has $m$ edges, $g$ interior lattice points and $n$ boundary
lattice points, solving a conjecture of Cr\'etois and Lang. Additionally, we
use abstract tropical curves to construct a compactification of this moduli
space by adding points that correspond to collections of curves that can be
patchworked together to produce a curve in $\mathcal{H}_\Delta$. This
compactification comes with a natural stratification with the same poset as the
secondary polytope of $\Delta$.
