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 analog 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 an anomalous response can be taken as a working definition of a fractionalized strongly correlated analog of the Weyl semimetal state.