Theta characteristics of tropical $K_4$-curves

A $K_4$-curve is a smooth, proper curve X of genus 3 over a nonarchimedean field whose Berkovich skeleton $\mathrm{\Gamma }$ is a complete graph on 4 vertices. The curve X has 28 effective theta characteristics, i.e. the 28 bitangents to a canonical embedding, while $\mathrm{\Gamma }$ has exactly seven tropical theta characteristics, as shown by Zharkov. We prove that the 28 effective theta characteristics of a $K_4$-curve specialize to the theta characteristics of its minimal skeleton in seven groups of four.