How do close neutron star binaries form? Many aspects of our understanding of the complex channels through which compact binaries form remain uncertain. This talk focuses on a crucial phase in the formation of close binary pairs of neutron stars: a common envelope episode. During the common envelope phase, one star evolves to engulf its companion in a shared gaseous envelope. Before this envelope is ejected, drag forces pull the two stellar cores into a tighter orbit. But the neutron star can also accrete from the surrounding gas. This talk explores the hydrodynamics of the common envelope phase in order to model the metamorphosis of the neutron star and the binary orbit.