  We study the effect of baryonic processes on the shapes of dark matter (DM)
haloes from Illustris, a suite of hydrodynamical (Illustris) and DM-only
(Illustris-Dark) cosmological simulations performed with the moving-mesh code
{\sc arepo}. DM halo shapes are determined using an iterative method based on
the inertia tensor for a wide range of $z=0$ masses ($M_{200} = 1 \times
10^{11} - 3 \times 10^{14} M_\odot$). Convergence tests shows that the local DM
shape profiles are converged only for $r > 9\epsilon$, $\epsilon$ being the
Plummer-equivalent softening length, larger than expected. Haloes from
non-radiative simulations (i.e. neglecting radiative processes, star formation,
and feedback) exhibit no alteration in shapes from their DM-only counterparts:
thus moving-mesh hydrodynamics alone is insufficient to cause differences in DM
shapes. With the full galaxy-physics implementation, condensation of baryons
results in significantly rounder and more oblate haloes, with the median
minor-to-major axis ratio $\left< s\equiv c/a \right> \approx 0.7$, almost
invariant throughout the halo and across halo masses. This somewhat improves
the agreement between simulation predictions and observational estimates of the
Milky Way halo shape. Consistently, the velocity anisotropy of DM is also
reduced in Illustris, across halo masses and radii. Within the inner halo
($r=0.15 R_{200}$), both $s$ and $q$ (intermediate-to-major axis ratio) exhibit
non-monotonicity with galaxy mass, peaking at $m_* \approx 10^{10.5-11}
M_\odot$, which we find is due to the strong dependence of inner halo shape
with galaxy formation efficiency. Baryons in Illustris affect the correlation
of halo shape with halo properties, leading to a positive correlation of
sphericity of MW-mass haloes with halo formation time and concentration, the
latter being mildly more pronounced than in Illustris-Dark.
