Let $K$ denote a compact real symmetric subset of $\mathbb{C}$ and let $A_{\mathbb R}(K)$ 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 $(f,g)$ in $A_{\mathbb R}(K)^2$ 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 $A(K)$ is $1$. Finally, we also characterize all compact real symmetric sets $K$ such that $A_{\mathbb R}(K)$, respectively $C_{\mathbb R}(K)$, has Bass stable rank $1$.

We give a description of the monoid of Murray-von Neumann equivalence classes of projections for multiplier algebras of a wide class of $\sigma$-unital simple $C^\ast$-algebras $A$ with real rank zero and stable rank one. The lattice of ideals of this monoid, which is known to be crucial for understanding the ideal structure of the multiplier algebra $\mul$, is therefore analyzed. In important cases it is shown that, if $A$ has finite scale then the quotient of $\mul$ modulo any closed ideal $I$ that properly contains $A$ has stable rank one. The intricacy of the ideal structure of $\mul$ is reflected in the fact that $\mul$ can have uncountably many different quotients, each one having uncountably many closed ideals forming a chain with respect to inclusion.