  We explore extensions of tropical methods to arithmetic enumerative problems
such as $\mathbb{A}^1$-enumeration with values in the Grothendieck-Witt ring,
and rationality over Henselian valued fields, using bitangents to plane
quartics as a test case. We consider quartic curves over valued fields whose
tropicalizations are smooth and satisfy a mild genericity condition. We then
express obstructions to rationality of bitangents and their points of tangency
in terms of twisting of edges of the tropicalization; the latter depends only
on the tropicalization and the initial coefficients of the defining equation
modulo squares. We also show that the GW-multiplicity of a tropical bitangent,
i.e., the multiplicity with which its lifts contribute to the
$\mathbb{A}^1$-enumeration of bitangents as defined by Larson and Vogt, can be
computed from the tropicalization of the quartic together with the initial
coefficients of the defining equation. As an application, we show that the four
lifts of most tropical bitangent classes contribute $2\mathbb{H}$, twice the
class of the hyperbolic plane, to the $\mathbb{A}^1$-enumeration. These results
rely on a degeneration theorem relating the Grothendieck-Witt ring of a
Henselian valued field to the Grothendieck-Witt ring of its residue field, in
residue characteristic not equal to two.
