  Recent work of Kass--Wickelgren gives an enriched count of the $27$ lines on
a smooth cubic surface over arbitrary fields. Their approach using
$\mathbb{A}^1$-enumerative geometry suggests that other classical enumerative
problems should have similar enrichments, when the answer is computed as the
degree of the Euler class of a relatively orientable vector bundle. Here, we
consider the closely related problem of the $28$ bitangents to a smooth plane
quartic. However, it turns out the relevant vector bundle is not relatively
orientable and new ideas are needed to produce enriched counts. We introduce a
fixed "line at infinity," which leads to enriched counts of bitangents that
depend on their geometry relative to the quartic and this distinguished line.
