  Interaction induced topological states remain one of the fascinating phases
in condensed matter physics. The exciton condensate has recently sparked
renewed interest due to the discovery of new candidate materials and its
driving force to realize exotic topological states. In this work, we explore
the exciton orders induced high-order topology in the bilayer quantum spin Hall
insulators and find that the topological excitonic corner states can be
realized by tuning the gate and magnetic field. When an in-plane Zeeman field
is applied to the system, two or four excitonic boundary-obstructed corner
states emerge in the bilayer system for distinct possible $s$-wave excitonic
pairings. Besides, we also find a two-dimensional excitonic Weyl nodal phase,
which supports flat band edge states connecting the bulk Weyl nodes.
