  Extensions of the standard models of particle physics and cosmology often
lead to long-range fifth forces with properties dependent on gravitational
environment. Fifth forces on astrophysical scales are best studied in the
cosmic web where perturbation theory breaks down. We present constraints on
chameleon- and symmetron-screened fifth forces with Yukawa coupling and
megaparsec range -- as well as unscreened fifth forces with differential
coupling to galactic mass components -- by searching for the displacements they
predict between galaxies' stars and gas. Taking data from the Alfalfa HI
survey, identifying galaxies' gravitational environments with the maps of
Desmond et al. (2018a) and forward-modelling with a Bayesian likelihood
framework, we set upper bounds on fifth-force strength relative to Newtonian
gravity from $\Delta G/G_N < \text{few} \: \times 10^{-4}$ for range $\lambda_C
= 50$ Mpc, to $\Delta G/G_N \lesssim 0.1$ for $\lambda_C = 500$ kpc. In $f(R)$
gravity this requires $f_{R0} < \text{few} \: \times \: 10^{-8}$. The analogous
bounds without screening are $\Delta G/G_N < \text{few} \: \times 10^{-4}$ and
$\Delta G/G_N < \text{few} \times 10^{-3}$. These are the tightest and among
the only fifth-force constraints on galaxy scales. We show how our results may
be strengthened with future survey data and identify the key features of an
observational programme for furthering fifth-force tests beyond the Solar
System.
