  We present BAHAMAS-SIDM, the first large-volume, (400/h Mpc)^3, cosmological
simulations including both self-interacting dark matter (SIDM) and baryonic
physics. These simulations are important for two primary reasons: 1) they
include the effects of baryons on the dark matter distribution 2) the baryon
particles can be used to make mock observables that can be compared directly
with observations. As is well known, SIDM haloes are systematically less dense
in their centres, and rounder, than CDM haloes. Here we find that that these
changes are not reflected in the distribution of gas or stars within galaxy
clusters, or in their X-ray luminosities. However, gravitational lensing
observables can discriminate between DM models, and we present a menu of tests
that future surveys could use to measure the SIDM interaction strength. We
ray-trace our simulated galaxy clusters to produce strong lensing maps.
Including baryons boosts the lensing strength of clusters that produce no
critical curves in SIDM-only simulations. Comparing the Einstein radii of our
simulated clusters with those observed in the CLASH survey, we find that at
velocities around 1000 km/s an SIDM cross-section of sigma/m > 1 cm^2/g is
likely incompatible with observed cluster lensing.
