  Higher-order topological insulators (HOTIs) are a novel type of topological
phases which supports $d$-dimensional topological boundary states in
$D$-dimensional systems with $D-d>1$. In this work, we theoretically predict
that interlayer couplings in AB-stacked bilayer transition-metal
dichalcogenides (TMDs) lead to the emergence of extrinsic second-order
topological phases, where corner states are induced by the band inversion of
zigzag edge bands. We find that the systems feature a quantized multiband Berry
phase defined for a zigzag nanoribbon geometry, unveiling the nontrivial
topological properties of its two zigzag edges. With detailed investigation
into the bilayer TMDs under different geometries, we find two types of
boundary-obstructed corner states arising from different corner terminations of
either the same type of or heterogeneous zigzag edges. The topological nature
of these corner states and their degeneracy is further analyzed with both the
crystalline symmetries of different geometries, and a topological phase
transition of the Berry phase induced by a layer-dependent onsite energy.
