  Joint analyses of small-scale cosmological structure probes are relatively
unexplored and promise to advance measurements of microphysical dark matter
properties using heterogeneous data. Here, we present a multidimensional
analysis of dark matter substructure using strong gravitational lenses and the
Milky Way (MW) satellite galaxy population, accounting for degeneracies in
model predictions and using covariances in the constraining power of these
individual probes for the first time. We simultaneously infer the projected
subhalo number density and the half-mode mass describing the suppression of the
subhalo mass function in thermal relic warm dark matter (WDM),
$M_{\mathrm{hm}}$, using the semi-analytic model $\texttt{Galacticus}$ to
connect the subhalo population inferred from MW satellite observations to the
strong lensing host halo mass and redshift regime. Combining MW satellite and
strong lensing posteriors in this parameter space yields
$M_{\mathrm{hm}}<10^{7.0}\ \mathrm{M}_{\mathrm{\odot}}$ (WDM particle mass
$m_{\mathrm{WDM}}>9.7\ \mathrm{keV}$) at $95\%$ confidence and disfavors
$M_{\mathrm{hm}}=10^{7.4}\ \mathrm{M}_{\mathrm{\odot}}$ ($m_{\mathrm{WDM}}=7.4\
\mathrm{keV}$) with a 20:1 marginal likelihood ratio, improving limits on
$m_{\mathrm{WDM}}$ set by the two methods independently by $\sim 30\%$. These
results are marginalized over the line-of-sight contribution to the strong
lensing signal, the mass of the MW host halo, and the efficiency of subhalo
disruption due to baryons, and are robust to differences in the disruption
efficiency between the MW and strong lensing regimes at the $\sim 10\%$ level.
This work paves the way for unified analyses of next-generation small-scale
structure measurements covering a wide range of scales and redshifts.
