  We analyze theories in which a supersymmetric sector is coupled to a
supersymmetry-breaking sector described by a non-linear realization. We show
how to consistently couple N=1 supersymmetric matter to non-supersymmetric
matter in such a way that all interactions are invariant under non-linear
supersymmetry transformations. We extend this formalism to couple N=2
supersymmetric matter to N=1 superfields that lack N=2 partners but transform
in a non-linear representation of the N=2 algebra. In particular, we show how
to couple an N=2 vector to N=1 chiral fields in a consistent way. This has
important applications to effective field theories describing the interactions
of D-brane world-volume fields with bulk fields. We apply our method to study
systems where different sectors break different halves of supersymmetry, which
appear naturally in models of intersecting branes.
