  This paper studies non-crossing geometric perfect matchings. Two such perfect
matchings are \emph{compatible} if they have the same vertex set and their
union is also non-crossing. Our first result states that for any two perfect
matchings $M$ and $M'$ of the same set of $n$ points, for some $k\in\Oh{\log
n}$, there is a sequence of perfect matchings $M=M_0,M_1,...,M_k=M'$, such that
each $M_i$ is compatible with $M_{i+1}$. This improves the previous best bound
of $k\leq n-2$. We then study the conjecture: \emph{every perfect matching with
an even number of edges has an edge-disjoint compatible perfect matching}. We
introduce a sequence of stronger conjectures that imply this conjecture, and
prove the strongest of these conjectures in the case of perfect matchings that
consist of vertical and horizontal segments. Finally, we prove that every
perfect matching with $n$ edges has an edge-disjoint compatible matching with
approximately $4n/5$ edges.
