We introduce a semisimple tensor category $\mc{ O}^{ int}_q(m|n)$ of modules over an quantum ortho-symplectic superalgebra. It is a natural counterpart of the category of finitely dominated integrable modules over the quantum classical (super) algebra of type $B_{ m+n}$, $C_{ m+n}$, $D_{ m+n}$ or $B(0,m+n)$ from a viewpoint of super duality. We classify the irreducible modules in $\mc{ O}^{ int}_q(m|n)$ and show that an irreducible module in $\mc{ O}^{ int}_q(m|n)$ has a unique crystal base in case of type $B$ and $C$. An explicit description of the crystal graph is given in terms of a new combinatorial object called ortho-symplectic tableaux.
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