Cluster realization of positive representations of a split real quantum Borel subalgebra

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