  We characterize protostellar multiplicity in the Orion molecular clouds using
ALMA 0.87~mm and VLA 9~mm continuum surveys toward 328 protostars. These
observations are sensitive to projected spatial separations as small as
$\sim$20~au, and we consider source separations up to 10$^4$~au as potential
companions. The overall multiplicity fraction (MF) and companion fraction (CF)
for the Orion protostars are 0.30$\pm$0.03 and 0.44$\pm$0.03, respectively,
considering separations from 20 to 10$^4$~au. The MFs and CFs are corrected for
potential contamination by unassociated young stars using a probabilistic
scheme based on the surface density of young stars around each protostar. The
companion separation distribution as a whole is double peaked and inconsistent
with the separation distribution of solar-type field stars, while the
separation distribution of Flat Spectrum protostars is consistent solar-type
field stars. The multiplicity statistics and companion separation distributions
of the Perseus star-forming region are consistent with those of Orion. Based on
the observed peaks in the Class 0 separations at $\sim$100~au and
$\sim$10$^3$~au, we argue that multiples with separations $<$500~au are likely
produced by both disk fragmentation and turbulent fragmentation with migration,
and those at $\ga$10$^3$~au result primarily from turbulent fragmentation. We
also find that MFs/CFs may rise from Class 0 to Flat Spectrum protostars
between 100 and 10$^3$~au in regions of high YSO density. This finding may be
evidence for migration of companions from $>$10$^3$~au to $<$10$^3$~au, and
that some companions between 10$^3$ and 10$^4$~au must be (or become) unbound.
