  The formation and evolution of galaxies is known to be sensitive to tidal
processes leading to intrinsic correlations between their shapes and
orientations. Such correlations can be measured to high significance today,
suggesting that cosmological information can be extracted from them. Among the
most pressing questions in particle physics and cosmology is the nature of dark
matter. If dark matter is self-interacting, it can leave an imprint on galaxy
shapes. In this work, we investigate whether self-interactions can produce a
long-lasting imprint on intrinsic galaxy shape correlations. We investigate
this observable at low redshift ($z<0.4$) using a state-of-the-art suite of
cosmological hydro-dynamical simulations where the dark matter model is varied.
We find that dark matter self-interactions induce a mass dependent suppression
in the intrinsic alignment signal by up to 50\% out to ten's of mega-parsecs,
showing that self-interactions can impact structure outside the very core of
clusters. We find evidence that self-interactions have a scale-dependent impact
on the intrinsic alignment signal that is sufficiently different from
signatures introduced by differing baryonic physics prescriptions, suggesting
that it is detectable with up-coming all-sky surveys.
