Let K denote a compact real symmetric subset of C and let AR(K) denote the real Banach algebra of all real symmetric continuous functions on K which are analytic in the interior K of K, endowed with the supremum norm. We characterize all unimodular pairs (f, g) in AR(K) 2 which are reducible. In addition, for an arbitrary compact K in C, we give a new proof (not relying on Banach algebra theory...