  Astrometry from {\it Gaia} DR3 has enabled the discovery of a sample of 3000+
binaries containing white dwarfs (WD) and main-sequence (MS) stars in
relatively wide orbits, with orbital periods $P_{\rm orb} = (100-1000)$ d. This
population was not predicted by binary population synthesis models before {\it
Gaia} and -- if the {\it Gaia} orbits are robust -- likely requires very
efficient envelope ejection during common envelope evolution (CEE). To assess
the reliability of the {\it Gaia} solutions, we measured multi-epoch radial
velocities (RVs) of 31 WD+MS binary candidates with $P_{\rm orb} = (40-300)$ d
and \texttt{AstroSpectroSB1} orbital solutions. We jointly fit the RVs and
astrometry, allowing us to validate the {\it Gaia} solutions and tighten
constraints on component masses. We find a high success rate for the {\it Gaia}
solutions, with only 2 out of the 31 systems showing significant discrepancies
between their {\it Gaia} orbital solutions and our RVs. Joint fitting of RVs
and astrometry allows us to directly constrain the secondary-to-primary flux
ratio $\mathcal{S}$, and we find $\mathcal{S}\lesssim 0.02$ for most objects,
confirming the companions are indeed WDs. We tighten constraints on the
binaries' eccentricities, finding a median $e\approx 0.1$. These eccentricities
are much lower than those of normal MS+MS binaries at similar periods, but much
higher than predicted for binaries formed via stable mass transfer. We present
MESA single and binary evolution models to explore how the binaries may have
formed. The orbits of most binaries in the sample can be produced through CEE
that begins when the WD progenitor is an AGB star, corresponding to initial
separations of $2-5$ au. Roughly 50\% of all post-common envelope binaries are
predicted to have first interacted on the AGB, ending up in wide orbits like
these systems.
