  Using density-functional ab initio theoretical techniques, we study
(Ga$_{1-x}$In$_x$)$_2$O$_3$ in both its equilibrium structures (monoclinic
$\beta$ and bixbyite) and over the whole range of composition. We establish
that the alloy exhibits a large and temperature-independent miscibility gap. On
the low-$x$ side, the favored phase is isostructural with $\beta$-Ga$_2$O$_3$;
on the high-$x$ side, it is isostructural with bixbyite In$_2$O$_3$. The
miscibility gap opens between approximately 15\% and 55\% In content for the
bixbyite alloy grown epitaxially on In$_2$O$_3$, and 15\% and 85\% In content
for the free-standing bixbyite alloy. The gap, volume and band offsets to the
parent compound also exhibit anomalies as function of $x$. Specifically, the
offsets in epitaxial conditions are predominantly type-B staggered, but have
opposite signs in the two end-of-range phases.
