  The manifold of the fundamental domain of the Brillouin zone is always
considered to be a torus. However, under the synthetic gauge field, the
Brillouin manifold can be modified by the projective symmetries, resulting in
unprecedented topological properties. Here, we realize a real-projective-plane
hybrid-order topological insulator in a phononic crystal by introducing the Z_2
gauge field. Such insulator hosts two momentum-space non-symmorphic reflection
symmetries, which change the Brillouin manifold from a torus to a real
projective plane. These symmetries can simultaneously lead to Klein-bottle and
quadrupole topologies in different bulk gaps. The non-symmorphic reflection
symmetries on Brillouin real projective plane, edge states of Klein-bottle
insulator, and corner states of quadrupole insulator are observed. These
results evidence the hybrid-order topology on Brillouin manifold beyond the
torus, and enrich the topological physics.
