  For a real reflection group the reflecting hyperplanes cut out on the unit
sphere a simplicial complex called the Coxeter complex. Abramenko showed that
each reflecting hyperplane meets the Coxeter complex in another Coxeter complex
if and only if the Coxeter diagram contains no subdiagram of type $D_4$, $F_4$,
or $H_4$. The present paper extends Abramenko's result to a wider class of
complex reflection groups. These groups have a Coxeter-like presentation and a
Coxeter-like complex called the Milnor fiber complex. Our first main theorem
classifies the groups whose reflecting hyperplanes meet the Milnor fiber
complex in another Milnor fiber complex. To understand better the walls that
fail to be Milnor fiber complexes we introduce Milnor walls. Our second main
theorem generalizes Abramenko's result in a second way. It says that each wall
of a Milnor fiber complex is a Milnor wall if and only if the diagram contains
no subdiagram of type $D_4$, $F_4$, or $H_4$.
