  Galactic stellar haloes are largely composed of the remnants of galaxies
accreted during the assembly of their host galaxies, and hence their properties
reflect the mass spectrum and post-accretion evolution of their satellites. As
the nature of dark matter (DM) can affect both, we explore how the properties
of the accreted stellar component vary across cold (CDM), warm (WDM) and
self-interacting (SIDM) models. We do this by studying accreted stellar
populations around eight MW-mass haloes using cosmological hydrodynamical
simulations based on the EAGLE galaxy formation model, in which we find that
the accreted stellar mass remains similar across models. Contrary to WDM, which
only presents minor differences relative to CDM, the distribution of accreted
stars in SIDM changes significantly within $0.05R_{200}$ ($10\,\mathrm{kpc}$).
The central density reduces to $\langle \rho^{\mathrm{SIDM}}_{\mathrm{exsitu}}
/ \rho^{\mathrm{CDM}}_{\mathrm{exsitu}} \rangle = 0.3$ and has a shallower
radial dependence, with logarithmic density slopes of $\langle
\alpha_{\mathrm{SIDM}} \rangle = -1.4$ vs $\langle \alpha_{\mathrm{CDM}}
\rangle = -1.7$. Additionally, stars are on more tangential orbits than their
CDM counterparts, with a change in the velocity anisotropy of $\langle \Delta
\beta \rangle = - 0.2$. Finally, SIDM stellar haloes have the largest number
and prominence of overdensities in radius vs radial velocity space. This is due
to a combination of shorter stellar halo progenitor merging timescales and
shallower host potentials, with the former resulting in less time for dynamical
friction and radialisation to operate. In summary, we show that the phase-space
structure of Galactic stellar haloes encode key information that can be used to
distinguish and rule out different DM models.
