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Let 
$K$ denote a compact real symmetric subset of 
$\mathbb{C}$ and let 
${{A}_{\mathbb{R}}}\left( K \right)$ denote the real Banach algebra of all real symmetric continuous functions on $K$ that are analytic in the interior 
${{K}^{\circ }}$ of $K$, endowed with the supremum norm. We characterize all unimodular pairs 
$\left( f,\,g \right)$ in 
${{A}_{\mathbb{R}}}{{\left( K \right)}^{2}}$ which are reducible. In addition, for an arbitrary compact $K$ in $\mathbb{C}$, we give a new proof (not relying on Banach algebra theory or elementary stable rank techniques) of the fact that the Bass stable rank of 
$A\left( K \right)$ is 1. Finally, we also characterize all compact real symmetric sets $K$ such that ${{A}_{\mathbb{R}}}\left( K \right)$, respectively 
${{C}_{\mathbb{R}}}\left( K \right)$, has Bass stable rank 1.
