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.
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