  In this paper, complement-equivalent arithmetic Zariski pairs will be
exhibited answering in the negative a question by Eyral-Oka on these curves and
their groups. A complement-equivalent arithmetic Zariski pair is a pair of
complex projective plane curves having Galois-conjugate equations in some
number field whose complements are homeomorphic, but whose embeddings in
$\mathbb{P}^2$ are not.
  Most of the known invariants used to detect Zariski pairs depend on the
\'etale fundamental group. In the case of Galois-conjugate curves, their
\'etale fundamental groups coincide. Braid monodromy factorization appears to
be sensitive to the difference between \'etale fundamental groups and
homeomorphism class of embeddings.
