  We consider tensor products of finite-dimensional representations of a
coideal subalgebra in $U_{q}(\mathfrak{sl}_2)$. We present an explicit
expression for the dual of the canonical bases through a diagrammatic
presentation. We show that the decomposition of tensor products of dual
canonical bases and the action of the coideal subalgebra have integral and
positive properties. As an application, we consider the eigensystem of the
generator of the coideal subalgebra on the dual canonical bases. We provide all
the eigenvalues and obtain an explicit expression of the eigenfunction for the
largest eigenvalue. The sum of the components of this eigenfunction is
conjectured to be equal to the total number of arrangements of bishops with a
certain symmetry.
