  A long-standing prediction of quantum electrodynamics, yet to be
experimentally observed, is the interaction between real photons in vacuum. As
a consequence of this interaction, the vacuum is expected to become
birefringent and dichroic if a strong laser field polarizes its virtual
particle--antiparticle dipoles. Here, we derive how a generally polarized probe
photon beam is influenced by both vacuum birefringence and dichroism in a
strong linearly polarized plane-wave laser field. Furthermore, we consider an
experimental scheme to measure these effects in the nonperturbative high-energy
regime, where the Euler-Heisenberg approximation breaks down. By employing
circularly polarized high-energy probe photons, as opposed to the
conventionally considered linearly polarized ones, the feasibility of
quantitatively confirming the prediction of nonlinear QED for vacuum
birefringence at the $5\sigma$ confidence level on the time scale of a few days
is demonstrated for upcoming 10 PW laser systems. Finally, dichroism and
anomalous dispersion in vacuum are shown to be accessible at these facilities.
