  Context. The fraction of field binaries on retrograde orbits about the Milky
Way is significantly lower compared to its prograde counterpart. Chemical and
dynamical evidence suggests that the retrograde stellar population originates
from $\omega$ Centauri, which is either the most massive globular cluster (GC)
of the Milky Way or the putative core of a former dwarf galaxy. Aims. Star
formation conditions required to produce the retrograde binary population are
constrained assuming that the retrograde stellar population originates from
$\omega$ Centauri's progenitor. Methods. We match the observed low binary
fraction with dynamical population synthesis models, including a universal
initial binary population and dynamical processing in star clusters, making use
of the publicly available binary population synthesis tool BiPoS1. Results. It
is found that either the GC progenitor of $\omega$ Centauri must have formed
with a stellar density of $\approx 10^8 \; M_{sun} \; pc^{-3}$ or the $\omega$
Centauri dwarf galaxy's progenitor star cluster population must have formed in
an extreme starburst with a star formation rate exceeding $1000 \; M_{sun} \;
yr^{-1}$ and probably a top-heavy embedded-cluster mass function with
suppressed low-mass cluster formation. The separation and mass ratio
distribution for retrograde field binaries are predicted for comparison with
future observations. Conclusions. A viable solution for the deficiency of
binaries on retrograde orbits is presented, and star formation conditions for
$\omega$ Centauri as well as orbital parameter distributions for the Milky
Way's retrograde binary population are predicted. The dwarf galaxy origin for
$\omega$ Centauri is tentatively preferred within the present context.
