  A matching is compatible to two or more labeled point sets of size $n$ with
labels $\{1,\dots,n\}$ if its straight-line drawing on each of these point sets
is crossing-free. We study the maximum number of edges in a matching compatible
to two or more labeled point sets in general position in the plane. We show
that for any two labeled convex sets of $n$ points there exists a compatible
matching with $\lfloor \sqrt {2n}\rfloor$ edges. More generally, for any $\ell$
labeled point sets we construct compatible matchings of size
$\Omega(n^{1/\ell})$. As a corresponding upper bound, we use probabilistic
arguments to show that for any $\ell$ given sets of $n$ points there exists a
labeling of each set such that the largest compatible matching has
${\mathcal{O}}(n^{2/({\ell}+1)})$ edges. Finally, we show that $\Theta(\log n)$
copies of any set of $n$ points are necessary and sufficient for the existence
of a labeling such that any compatible matching consists only of a single edge.
