  We use the Cluster-EAGLE (C-EAGLE) hydrodynamical simulations to investigate
the effects of self-interacting dark matter (SIDM) on galaxies as they fall
into clusters. We find that SIDM galaxies follow similar orbits to their Cold
Dark Matter (CDM) counterparts, but end up with ${\sim}$25 per cent less mass
by the present day. One in three SIDM galaxies are entirely disrupted, compared
to one in five CDM galaxies. However, the excess stripping will be harder to
observe than suggested by previous DM-only simulations because the most
stripped galaxies form cores and also lose stars: the most discriminating
objects become unobservable. The best test will be to measure the
stellar-to-halo mass relation (SHMR) for galaxies with stellar mass
$10^{10-11}\,\mathrm{M}_{\odot}$. This is 8 times higher in a cluster than in
the field for a CDM universe, but 13 times higher for an SIDM universe. Given
intrinsic scatter in the SHMR, these models could be distinguished with
noise-free galaxy-galaxy strong lensing of ${\sim}32$ cluster galaxies.
