  Compositional asymmetry between the leaflets of bilayer membranes modifies
their phase behaviour, and is thought to influence other important features
such as mechanical properties and protein activity. We address here how phase
behaviour is affected by passive phospholipid \textit{flip-flop}, such that the
compositional asymmetry is not fixed. We predict transitions from "pre
flip-flop" behaviour to a restricted set of phase equilibria that can persist
in the presence of passive flip-flop. Surprisingly, such states are not
necessarily symmetric. We further account for external symmetry-breaking, such
as a preferential substrate interaction, and show how this can stabilise
strongly asymmetric equilibrium states. Our theory explains several
experimental observations of flip-flop mediated changes in phase behaviour, and
shows how domain formation and compositional asymmetry can be controlled in
concert, by manipulating passive flip-flop rates and applying external fields.
