The Cosmic Neutrino Background (CNB) encodes a wealth of information, but has not yet been
observed directly. To determine the prospects of detection and to study its information content,
we reconstruct the phase-space distribution of local relic neutrinos from the three-dimensional
distribution of matter within 200 h
-1 Mpc of the Milky Way. Our analysis relies on
constrained realization simulations and forward modelling of the 2M++ galaxy
catalogue. We find that the angular distribution of neutrinos is anti-correlated with the
projected matter density, due to the capture and deflection of neutrinos by massive structures
along the line of sight. Of relevance to tritium capture experiments, we find that the
gravitational clustering effect of the large-scale structure on the local number density of
neutrinos is more important than that of the Milky Way for neutrino masses less than
0.1 eV. Nevertheless, we predict that the density of relic neutrinos is close to the cosmic
average, with a suppression or enhancement over the mean of (-0.3%, +7%, +27%) for masses
of (0.01, 0.05, 0.1) eV. This implies no more than a marginal increase in the event rate for
tritium capture experiments like PTOLEMY. We also predict that the CNB and CMB rest frames
coincide for 0.01 eV neutrinos, but that neutrino velocities are significantly perturbed for
masses larger than 0.05 eV. Regardless of mass, we find that the angle between the neutrino
dipole and the ecliptic plane is small, implying a near-maximal annual modulation in the bulk
velocity. Along with this paper, we publicly release our simulation data, comprising more than
100 simulations for six different neutrino masses.