I will present BAHAMAS-SIDM, the first large-volume, cosmological simulations including both self-interacting dark matter (SIDM) and baryonic physics. As is well known, SIDM haloes are systematically less dense in their centres, and rounder, than CDM haloes. In our simulations, 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 I will show how the distribution of Einstein radii can be used to constrain the SIDM cross-section. I will then discuss how we used the offsets between brightest cluster galaxies and the DM haloes that host them to place one of the tightest constraints to date on DM self-interactions. Finally, I will discuss using these simulations to test analytic methods for modelling SIDM density profiles in the presence of baryons.