  We explore the predictions of Milgromian gravity (MOND) in the Local Universe
by considering the distribution of the `phantom' dark matter (PDM) that would
source the MOND gravitational field in Newtonian gravity, allowing an easy
comparison with the dark matter framework. For this, we specifically deal with
the quasi-linear version of MOND (QUMOND). We compute the
`stellar-to-(phantom)halo-mass relation' (SHMR), a monotonically increasing
power-law resembling the SHMR observationally deduced from spiral galaxy
rotation curves in the Newtonian context. We show that the
gas-to-(phantom)halo-mass relation is flat. We generate a map of the Local
Volume in QUMOND, highlighting the important influence of distant galaxy
clusters, in particular Virgo. This allows us to explore the scatter of the
SHMR and the average density of PDM around galaxies in the Local Volume,
$\Omega_{\rm pdm} \approx 0.1$, below the average cold dark matter density in a
$\Lambda$CDM Universe. We provide a model of the Milky Way in its external
field in the MOND context, which we compare to an observational estimate of the
escape velocity curve. Finally, we highlight the peculiar features related to
the external field effect in the form of negative PDM density zones in the
outskirts of each galaxy, and test a new analytic formula for computing galaxy
rotation curves in the presence of an external field in QUMOND. While we show
that the negative PDM density zones would be difficult to detect dynamically,
we quantify the weak lensing signal they could produce for lenses at $z \sim
0.3$.
