  The Bloch wave functions have been playing a crucial role in the diagnosis of
topological phases in non-interacting systems. However, the Bloch waves are no
longer applicable in the presence of finite Coulomb interaction and alternative
approaches are needed to identify the topological indices. In this paper, we
focus on three-dimensional higher-order topological insulators protected by
$C_4T$ symmetry and show that the topological index can be computed through
eigenstates of inverse Green's function at zero frequency. If there is an
additional $S_4$ rotoinversion symmetry, the topological index $P_3$ can be
determined by eigenvalues of $S_4$ at high symmetry momenta, similar to the
Fu-Kane parity criterion. We verify this method using many-body exact
diagonalization in higher-order topological insulators with interaction. We
also discuss the realization of this higher-order topological phase in
tetragonal lattice structure with $C_4T$-preserving magnetic order. Finally, we
discuss the boundary conditions necessary for the hinge states to emerge and
show that these hinge states exist even when the boundary is smooth and without
a sharp hinge.
