  We present shape measurements of Milky Way-sized dark matter halos at
redshift $z=0$ in a suite of 30 zoom simulations from the Auriga project. We
compare the results in full magnetohydrodynamics against dark matter only
simulations and find a strong influence of baryons in making dark matter haloes
rounder at all radii compared to their dark matter only counterparts. At
distances $\lesssim 30$ kpc, rounder dark matter distributions correlate with
extended massive stellar discs and low core gas densities. We measure the
alignment between the halo and the disc shapes at different radii and find a
high degree of alignment at all radii for most of the galaxies. In some cases
the alignment significantly changes as a function of radius implying that the
halo shape twists; this effect correlates with recently formed bulges and is
almost absent in the dark matter only simulations. In a comparison against
observational constraints we find that $20\%$ of halos in our sample are
consistent with observational results derived from the Pal 5 stream that
favours an almost spherical shape. Including baryons is a required element to
achieve this level of agreement. In contrast, none of the simulations (neither
dark matter only nor with baryons) match the constraints derived from the
Sagittarius stream that favour an oblate dark matter halo.
