  The possibility of realizing non-Abelian statistics and utilizing it for
topological quantum computation (TQC) has generated widespread interest.
However, the non-Abelian statistics that can be realized in most accessible
proposals is not powerful enough for universal TQC. In this paper, we consider
a simple bilayer fractional quantum Hall (FQH) system with the $1/3$ Laughlin
state in each layer. We show that interlayer tunneling can drive a transition
to an exotic non-Abelian state that contains the famous `Fibonacci'anyon, whose
non-Abelian statistics is powerful enough for universal TQC. Our analysis rests
on startling agreements from a variety of distinct methods, including thin
torus limits, effective field theories, and coupled wire constructions. We
provide evidence that the transition can be continuous, at which point the
charge gap remains open while the neutral gap closes. This raises the question
of whether these exotic phases may have already been realized at $\nu = 2/3$ in
bilayers, as past experiments may not have definitively ruled them out.
