  We introduce a notion of compatibility for multiplicity matrices. This gives
rise to a necessary condition for the join of two (possibly disconnected)
graphs $G$ and $H$ to be the pattern of an orthogonal symmetric matrix, or
equivalently, for the minimum number of distinct eigenvalues $q$ of $G\vee H$
to be equal to two. Under additional hypotheses, we show that this necessary
condition is also sufficient. As an application, we prove that $q(G\vee H)$ is
either two or three when $G$ and $H$ are unions of complete graphs, and we
characterise when each case occurs.
