  Let $\mathcal{O}^{int}_q(m|n)$ be a semisimple tensor category of modules
over a quantum ortho-symplectic superalgebra of type $B, C, D$ introduced in
the author's previous work. It is a natural counterpart of the category of
finitely dominated integrable modules over a quantum group of type $B, C, D$
from a viewpoint of super duality. Continuing the previous work on type $B$ and
$C$, we classify the irreducible modules in $\mathcal{O}^{int}_q(m|n)$, and
prove the existence and uniqueness of their crystal bases in case of type $D$.
A new combinatorial model of classical crystals of type $D$ is introduced,
whose super analogue gives a realization of crystals for the highest weight
modules in $\mathcal{O}^{int}_q(m|n)$.
