  Theories of modified gravity generically violate the strong equivalence
principle, so that the internal dynamics of a self-gravitating system in free
fall depends on the strength of the external gravitational field (the external
field effect). We fit rotation curves (RCs) from the SPARC database with a
model inspired by Milgromian dynamics (MOND), which relates the outer shape of
a RC to the external Newtonian field from the large-scale baryonic matter
distribution through a dimensionless parameter $e_{\rm N}$. We obtain a
$>4\sigma$ statistical detection of the external field effect (i.e. $e_{\rm
N}>0$ on average), confirming previous results. We then locate the SPARC
galaxies in the cosmic web of the nearby Universe and find a striking contrast
in the fitted $e_{\rm N}$ {values} for galaxies in underdense versus overdense
regions. Galaxies in an underdense region between 22 and 45 Mpc from the
celestial axis in the northern sky have RC fits consistent with $e_{\rm
N}\simeq0$, while those in overdense regions adjacent to the CfA2 great wall
and the Perseus-Pisces supercluster return $e_{\rm N}$ that are a factor of two
larger than the median for SPARC galaxies. We also calculate independent
estimates of $e_{\rm N}$ from galaxy survey data and find that they agree with
the $e_{\rm N}$ inferred from the RCs within the uncertainties, the chief
uncertainty being the spatial distribution of baryons not contained in galaxies
or clusters.
