  The anticanonical complex generalizes the Fano polytope from toric geometry
and has been used to study Fano varieties with torus action so far. We work out
the case of complete intersections in toric varieties defined by non-degenerate
systems of Laurent polynomials. As an application, we classify the terminal
Fano threefolds that are embedded into a fake weighted projective space via a
general system of Laurent polynomials.
