  Comparison between multidimensional persistent Betti numbers is often based
on the multidimensional matching distance. While this metric is rather simple
to define and compute by considering a suitable family of filtering functions
associated with lines having a positive slope, it has two main drawbacks.
First, it forgets the natural link between the homological properties of
filtrations associated with lines that are close to each other. As a
consequence, part of the interesting homological information is lost. Second,
its intrinsically discontinuous definition makes it difficult to study its
properties. In this paper we introduce a new matching distance for 2D
persistent Betti numbers, called coherent matching distance and based on
matchings that change coherently with the filtrations we take into account. Its
definition is not trivial, as it must face the presence of monodromy in
multidimensional persistence, i.e. the fact that different paths in the space
parameterizing the above filtrations can induce different matchings between the
associated persistent diagrams. In our paper we prove that the coherent 2D
matching distance is well-defined and stable.
