  Brodsky, Joswig, Morrison and Sturmfels showed that not all abstract tropical
curves of genus $3$ can be realized as a tropicalization of a quartic in the
euclidean plane. In this article, we focus on the interior of the maximal cones
in the moduli space and classify all curves which can be realized as a faithful
tropicalization in a tropical plane. Reflecting the algebro-geometric world, we
show that these are exactly those which are not realizably hyperelliptic.
  Our approach is constructive: For any not realizably hyperelliptic curve, we
explicitly construct a realizable model of the tropical plane and a faithfully
tropicalized quartic in it. These constructions rely on modifications resp.
tropical refinements. Conversely, we prove that any realizably hyperelliptic
curve cannot be embedded in such a fashion. For that, we rely on the theory of
tropical divisors and embeddings from linear systems, and recent advances in
the realizability of sections of the tropical canonical divisor.
