  We study the representation theory of a quantum symmetric pair
$(\mathbf{U},\mathbf{U}^{\jmath})$ with two parameters $p,q$ of type AIII, by
using highest weight theory and a variant of Kashiwara's crystal basis theory.
Namely, we classify the irreducible $\mathbf{U}^{\jmath}$-modules in a suitable
category and associate with each of them a basis at $p=q=0$, the
$\jmath$-crystal basis. The $\jmath$-crystal bases have nice combinatorial
properties as the ordinary crystal bases do.
