  Using a combination of deep 574ks Chandra data, XMM-Newton high-resolution
spectra, and optical Halpha+NII images, we study the nature and spatial
distribution of the multiphase plasma in M87. Our results provide direct
observational evidence of `radio mode' AGN feedback in action, stripping the
central galaxy of its lowest entropy gas and preventing star-formation. This
low entropy gas was entrained with and uplifted by the buoyantly rising
relativistic plasma, forming long "arms". These arms are likely oriented within
15-30 degrees of our line-of-sight. The mass of the uplifted gas in the arms is
comparable to the gas mass in the approximately spherically symmetric 3.8 kpc
core, demonstrating that the AGN has a profound effect on its immediate
surroundings. The coolest X-ray emitting gas in M87 has a temperature of ~0.5
keV and is spatially coincident with Halpha+NII nebulae, forming a multiphase
medium where the cooler gas phases are arranged in magnetized filaments. We
place strong upper limits of 0.06 Msun/yr on the amount of plasma cooling
radiatively from 0.5 keV and show that a uniform, volume-averaged heating
mechanism could not be preventing the cool gas from further cooling. All of the
bright Halpha filaments appear in the downstream region of the <3 Myr old shock
front, at smaller radii than ~0.6'. We suggest that shocks induce shearing
around the filaments, thereby promoting mixing of the cold gas with the ambient
hot ICM via instabilities. By bringing hot thermal particles into contact with
the cool, line-emitting gas, mixing can supply the power and ionizing particles
needed to explain the observed optical spectra. Mixing of the coolest X-ray
emitting plasma with the cold optical line emitting filamentary gas promotes
efficient conduction between the two phases, allowing non-radiative cooling
which could explain the lack of X-ray gas with temperatures under 0.5 keV.
