  We introduce a model of interacting Majorana fermions that describes a
superconducting phase with a topological order characterized by the Fibonacci
topological field theory. Our theory, which is based on a $SO(7)_1/(G_2)_1$
coset factorization, leads to a solvable one dimensional model that is extended
to two dimensions using a network construction. In addition to providing a
description of the Fibonacci phase without parafermions, our theory predicts a
closely related "anti-Fibonacci" phase, whose topological order is
characterized by the tricritical Ising model. We show that Majorana fermions
can split into a pair of Fibonacci anyons, and propose an interferometer that
generalizes the $Z_2$ Majorana interferometer and directly probes the Fibonacci
non-Abelian statistics.
