  The coupled-wires approach has been shown to be useful in describing
two-dimensional strongly interacting topological phases. In this manuscript we
extend this approach to three-dimensions, and construct a model for a
fractional strong topological insulator. This topologically ordered phase has
an exotic gapless state on the surface, called a fractional Dirac liquid, which
cannot be described by the Dirac theory of free fermions. Like in
non-interacting strong topological insulators, the surface is protected by the
presence of time-reversal symmetry and charge conservation. We show that upon
breaking these symmetries, the gapped fractional Dirac liquid presents unique
features. In particular, the gapped phase that results from breaking
time-reversal symmetry has a halved fractional Hall conductance of the form
$\sigma_{xy}=\frac{1}{2}\frac{e^{2}}{mh}$ if the filling is $\nu=1/m$. On the
other hand, if the surface is gapped by proximity coupling to an $s$-wave
superconductor, we end up with an exotic topological superconductor. To reveal
the topological nature of this superconducting phase, we partition the surface
into two regions: one with broken time-reversal symmetry and another coupled to
a superconductor. We find a fractional Majorana mode, which cannot be described
by a free Majorana theory, on the boundary between the two regions. The density
of states associated with tunneling into this one-dimensional channel is
proportional to $\omega^{m-1}$, in analogy to the edge of the corresponding
Laughlin state.
