  Decaying or annihilating dark matter particles could be detected through
gamma-ray emission from the species they decay or annihilate into. This is
usually done by modelling the flux from specific dark matter-rich objects such
as the Milky Way halo, Local Group dwarfs, and nearby groups. However, these
objects are expected to have significant emission from baryonic processes as
well, and the analyses discard gamma-ray data over most of the sky. Here we
construct full-sky templates for gamma-ray flux from the large-scale structure
within $\sim$200 Mpc by means of a suite of constrained $N$-body simulations
(CSiBORG) produced using the Bayesian Origin Reconstruction from Galaxies
algorithm. Marginalising over uncertainties in this reconstruction, small-scale
structure, and parameters describing astrophysical contributions to the
observed gamma-ray sky, we compare to observations from the Fermi Large Area
Telescope to constrain dark matter annihilation cross sections and decay rates
through a Markov Chain Monte Carlo analysis. We rule out the thermal relic
cross section for $s$-wave annihilation for all $m_\chi \lesssim 7 {\rm \,
GeV}/c^2$ at 95\% confidence if the annihilation produces gluons or quarks less
massive than the bottom quark. We infer a contribution to the gamma-ray sky
with the same spatial distribution as dark matter decay at $3.3\sigma$.
Although this could be due to dark matter decay via these channels with a decay
rate $\Gamma \approx 6 \times 10^{-28} {\rm \, s^{-1}}$, we find that a
power-law spectrum of index $p=-2.75^{+0.71}_{-0.46}$, likely of baryonic
origin, is preferred by the data.
