  The width of the $\omega$ meson in cold nuclear matter is computed in a
hadronic many-body approach, focusing on a detailed treatment of the medium
modifications of intermediate $\pi\rho$ states. The $\pi$ and $\rho$
propagators are dressed by their selfenergies in nuclear matter taken from
previously constrained many-body calculations. The pion selfenergy includes
$Nh$ and $\Delta h$ excitations with short-range correlations, while the $\rho$
selfenergy incorporates the same dressing of its $2\pi$ cloud with a full
3-momentum dependence and vertex corrections, as well as direct resonance-hole
excitations; both contributions were quantitatively fit to total
photo-absorption spectra and $\pi N\to\rho N$ scattering. Our calculations
account for in-medium decays of type $\omega N\to \pi N^{(*)}, \pi\pi
N(\Delta)$, and 2-body absorptions $\omega NN\to NN^{(*)},\pi NN$. This causes
deviations of the in-medium $\omega$ width from a linear behavior in density,
with important contributions from spacelike $\rho$ propagators. The $\omega$
width from the $\rho\pi$ cloud may reach up to 200 MeV at normal nuclear matter
density, with a moderate 3-momentum dependence. This largely resolves the
discrepancy of linear $T$-$\varrho$ approximations with the values deduced from
nuclear photoproduction measurements.
