Within condensed-matter systems, strong electronic interactions often lead to exotic quantum phases. A recent manifestation of this is the unexpected observation of magnetic quantum oscillations [1][2][3][4] and metallic thermal transport [5,6], both properties of systems with Fermi surfaces of itinerant quasiparticles, in the Kondo insulators SmB 6 and YbB 12 . To understand these phenomena, it is informative to study their evolution as the energy gap of the Kondo-Insulator state is closed by a large magnetic field. We show here that both the quantumoscillation frequency and the cyclotron mass display a strong field dependence in the resulting high-field metallic state in YbB 12 . By tracking the Fermi-surface area, we conclude that the same quasiparticle band gives rise to the quantum oscillations in both insulating and metallic states. These data are understood most simply using a twofluid picture where unusual quasiparticles, contributing little or nothing to charge transport, coexist with conventional fermions. In the metallic state this leads to a heavyfermion bad metal with negligible magnetoresistance, relatively high resistivity and a very large Kadowaki-Woods ratio, underlining the exotic nature of the fermion ensemble inhabiting YbB 12 .In Kondo insulators (KIs), an energy gap is opened up by strong coupling between a lattice of localized moments and the extended electronic states. The resulting Kondo gap E g is usually narrow (typically E g ≃ 5 − 20 meV), yet the rôle it plays in charge transport is more complicated than that of the bandgap in conventional semiconductors. A low-temperature (T ) saturation of the resistivity ρ has long been known in two prototypical KIs, SmB 6 and YbB 12 [7,8]; both are mixedvalence compounds with strong f − d hybridization that defines the band structure close to the Fermi energy. While the saturation might suggest additional metallic conduction channels, the high resistivity value within the weakly T -dependent "plateau" implies an unconventional nature for such channels [4,7].Recently, magnetic quantum oscillations, suggestive of a Fermi surface (FS), and thus totally unexpected in an insulator, have been detected in both SmB 6 and YbB 12 [1][2][3][4].