It has recently been discovered that the Harnack curves and their complexifications possess many other extremal properties. For instance, the amoeba of a complex Harnack curve is of maximal area, and it has the maximal number of "holes" that precisely correspond to the ovals of the real curve. In fact, the area of the holes can be taken as coordinates for the moduli space of Harnack curves of a given degree.
In the work of Andrei Okounkov and his collaborators the very same Harnack curves and their amoebas unexpectedly show up in combinatorial random surface models for partially dissolved crystals.