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We explore the predictions of Milgromian gravity (MOND) in the Local Universeby considering the distribution of the `phantom' dark matter (PDM) that wouldsource the MOND gravitational field in Newtonian gravity, allowing an easycomparison with the dark matter framework. For this, we specifically deal withthe quasi-linear version of MOND (QUMOND). We compute the`stellar-to-(phantom)halo-mass relation' (SHMR), a monotonically increasingpower-law resembling the SHMR observationally deduced from spiral galaxyrotation curves in the Newtonian context. We show that thegas-to-(phantom)halo-mass relation is flat. We generate a map of the LocalVolume in QUMOND, highlighting the important influence of distant galaxyclusters, in particular Virgo. This allows us to explore the scatter of theSHMR 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 externalfield in the MOND context, which we compare to an observational estimate of theescape velocity curve. Finally, we highlight the peculiar features related tothe external field effect in the form of negative PDM density zones in theoutskirts of each galaxy, and test a new analytic formula for computing galaxyrotation curves in the presence of an external field in QUMOND. While we showthat the negative PDM density zones would be difficult to detect dynamically,we quantify the weak lensing signal they could produce for lenses at $z \sim0.3$.

The phantom dark matter halos of the Local Volume in the context of modified Newtonian dynamics