  A model of two 1D ideal Bose gases A and B with strong AB attractions induced
by a p-wave AB Feshbach resonance is studied. The model is solved exactly by a
Bose-Bose duality mapping, and it is shown that there is no A-component or
B-component Bose-Einstein condensation and no AB-pair off-diagonal long-range
order (ODLRO), but both AA-pair and BB-pair ODLRO. After generalization by
adding even-wave AA and BB repulsion and reducing the strength of the odd-wave
AB attraction by Feshbach resonance detuning, a quantum phase transition occurs
between a phase with AB contact nodes and one with no such nodes.
