  Self-interacting dark matter (SIDM) models offer one way to reconcile
inconsistencies between observations and predictions from collisionless cold
dark matter (CDM) models on dwarf-galaxy scales. In order to incorporate the
effects of both baryonic and SIDM interactions, we study a suite of
cosmological-baryonic simulations of Milky-Way (MW)-mass galaxies from the
Feedback in Realistic Environments (FIRE-2) project where we vary the SIDM
self-interaction cross-section $\sigma/m$. We compare the shape of the main
dark matter (DM) halo at redshift $z=0$ predicted by SIDM simulations (at
$\sigma/m=0.1$, $1$, and $10$ cm$^2$ g$^{-1}$) with CDM simulations using the
same initial conditions. In the presence of baryonic feedback effects, we find
that SIDM models do not produce the large differences in the inner structure of
MW-mass galaxies predicted by SIDM-only models. However, we do find that the
radius where the shape of the total mass distribution begins to differ from
that of the stellar mass distribution is dependent on $\sigma/m$. This
transition could potentially be used to set limits on the SIDM cross-section in
the MW.
