  We show how background field inhomogeneities modify the non-perturbative
structure of the effective action. The simple Borel poles of the
Euler-Heisenberg effective action become branch points, and new branch points
also appear, indicating new non-perturbative effects. This information is
resurgently encoded in the perturbative weak field expansion, and becomes
physically significant for strongly inhomogeneous fields. We also show that
resurgent extrapolation methods permit the decoding of a surprising amount of
non-perturbative information from a relatively modest amount of perturbative
input, enabling accurate analytic continuations from weak field to strong
field, and of a spatially dependent magnetic background to a time dependent
electric background. These extrapolations are far superior to standard WKB and
locally constant field approximations.
