  We investigate a fundamental nonlinear process of vacuum photon emission in
the presence of strong electromagnetic fields going beyond the locally-constant
field approximation (LCFA), i.e., providing the exact treatment of the
spatiotemporal inhomogeneities of the external field. We examine a standing
electromagnetic wave formed by high-intensity laser pulses and benchmark the
approximate predictions against the results obtained by means of a precise
approach evaluating both the tadpole (reducible) and vertex (irreducible)
contributions. It is demonstrated that the previously used approximate methods
may fail to properly describe the quantitative characteristics of each of the
two terms. In the case of the tadpole contribution, the LCFA considerably
underestimates the number of photons emitted for sufficiently high frequency of
the external field. The vertex term predicts emission of a great number of soft
photons whose spectrum is no longer isotropic in contrast to the LCFA results.
A notable difference among the photon yields along different spatial
directions, which is not captured by the LCFA, represents an important
signature for experimental studies of the photon emission process. Since this
feature takes place unless the Keldysh parameter is much larger than unity, it
can also be used in indirect observation of the Schwinger mechanism.
