  We study high-field magnon dynamics and examine the dynamical structure
factor in the quasi-2D tetragonal Heisenberg antiferromagnet with interlayer
coupling corresponding to realistic materials. Within spin-wave theory, we show
that a non-zero interlayer coupling mitigates singular corrections to the
excitation spectrum occurring in the high-field regime that would otherwise
require a self-consistent approach beyond the 1/S approximation. For the fields
between the threshold for decays and saturation field we observe widening of
the two-magnon sidebands with significant shifting of the spectral weight away
from the quasiparticle peak. We find spectrum broadening throughout large
regions of the Brillouin zone, dramatic redistributions of spectral weight to
the two-magnon continuum, two-peak structures and other features clearly unlike
conventional single-particle peaks.
