  In our previous work, we studied the positive representations of split real
quantum groups $\mathcal{U}_{q\tilde{q}}(\mathfrak{g}_\mathbb{R})$ restricted
to its Borel part, and showed that they are closed under taking tensor
products. However, the tensor product decomposition was only constructed
abstractly using the GNS-representation of a $C^*$-algebraic version of the
Drinfeld-Jimbo quantum groups. In this paper, using the recently discovered
cluster realization of quantum groups, we write down the decomposition
explicitly by realizing it as a sequence of cluster mutations in the
corresponding quiver diagram representing the tensor product.
