  Recent studies have presented evidence for tension between the constraints on
Omega_m and sigma_8 from the cosmic microwave background (CMB) and measurements
of large-scale structure (LSS). This tension can potentially be resolved by
appealing to extensions of the standard model of cosmology and/or untreated
systematic errors in the modelling of LSS, of which baryonic physics has been
frequently suggested. We revisit this tension using, for the first time,
carefully-calibrated cosmological hydrodynamical simulations, which thus
capture the back reaction of the baryons on the total matter distribution. We
have extended the BAHAMAS simulations to include a treatment of massive
neutrinos, which currently represents the best motivated extension to the
standard model. We make synthetic thermal Sunyaev-Zel'dovich effect, weak
galaxy lensing, and CMB lensing maps and compare to observed auto- and
cross-power spectra from a wide range of recent observational surveys. We
conclude that: i) in general there is tension between the primary CMB and LSS
when adopting the standard model with minimal neutrino mass; ii) after
calibrating feedback processes to match the gas fractions of clusters, the
remaining uncertainties in the baryonic physics modelling are insufficient to
reconcile this tension; and iii) if one accounts for internal tensions in the
Planck CMB dataset (by allowing the lensing amplitude, A_Lens, to vary),
invoking a non-minimal neutrino mass, typically of 0.2-0.4 eV, can resolve the
tension. This solution is fully consistent with separate constraints from the
primary CMB and baryon acoustic oscillations.
