  Bilayer Kondo systems present interesting models to illustrate the
competition between the Kondo effect and intermoment exchange. Such bilayers
can exhibit two sharply distinct Fermi liquid phases which are distinguished by
whether or not the local moments participate in the Fermi sea. We study these
phases and the evolution from one to the other upon changing Kondo coupling. We
argue that an ordered state with spontaneous interlayer phase coherence
generically intervenes between the two Fermi liquids. Such a condensate phase
breaks a U(1) symmetry and is bounded by a finite-temperature
Kosterlitz-Thouless transition. Based on general arguments and mean-field
calculations we investigate the phase diagram and associated quantum phase
transitions.
