We implement two-cover descent for plane quartics over
with
all 28 bitangents rational and show that on a significant collection of test cases, it
resolves the existence of rational points. We also review a classical description of the
relevant moduli space and use it to generate examples. We observe that
local obstructions are quite rare for such curves and only seem to occur in
practice at primes of good reduction. In particular, having good reduction at
implies having no rational points. We also gather numerical data on two-Selmer ranks
of Jacobians of these curves, providing evidence these behave differently from those of
general abelian varieties due to the frequent presence of an everywhere locally trivial
torsor.
