  Formulating consistent theories describing strongly correlated metallic
topological phases is an outstanding problem in condensed matter physics. In
this work we derive a theory defining a fractionalized analogue of the Weyl
semimetal state: the fractional chiral metal. Our approach is to construct a
4+1D quantum Hall insulator by stacking 3+1D Weyl semimetals in a magnetic
field. In a strong enough field the low-energy physics is determined by the
lowest Landau level of each Weyl semimetal, which is highly degenerate and
chiral, motivating us to use a coupled-wire approach. The one-dimensional
dispersion of the lowest Landau level allows us to model the system as a set of
degenerate 1+1D quantum wires that can be bosonized in the presence of
electron-electron interactions and coupled such that a gapped phase is
obtained, whose response to an electromagnetic field is given in terms of a
Chern-Simons field theory. At the boundary of this phase we obtain the field
theory of a 3+1D gapless fractional chiral state, which we show is consistent
with a previous theory for the surface of a 4+1D Chern-Simons theory. The
boundary's response to an external electromagnetic field is determined by a
chiral anomaly with a fractionalized coefficient. We suggest that such
anomalous response can be taken as a working definition of a fractionalized
strongly correlated analogue of the Weyl semimetal state.
