  We use the reflection group trick to glue manifolds with corners that are
Borel-Serre compactifications of locally symmetric spaces of noncompact type
and obtain aspherical manifolds. We call these \emph{piecewise locally
symmetric} manifolds. This class of spaces provide new examples of aspherical
manifolds whose fundamental groups have the structure of a complex of groups.
These manifolds typically do not admit a locally $\CAT(0)$ metric. We prove
that any self homotopy equivalence of such manifolds is homotopic to a
homeomorphism. We compute the group of self homotopy equivalences of such a
manifold and show that it can contain a normal free abelian subgroup, and thus
can be infinite.
