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 that 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.