  We introduce the notion of locally consistent system of half-spaces for a
real hyperplane arrangement. We embed a sphere in the complexified complement
by shifting the real unit sphere into the imaginary direction indicated by the
half-spaces. We then prove that the sphere is homotopically trivial if and only
if the system of half-spaces is globally consistent. To prove its
non-triviality, we compute the twisted intersection number of the sphere with a
specific, explicitly constructed twisted Borel-Moore cycle.
