  In this paper we consider a two component system of coupled non linear
Schr\"odinger equations modeling the phase separation in the binary mixture of
Bose-Einstein condensates and other related problems. Assuming the existence of
solutions in the limit of large interspecies scattering length $\beta$ the
system reduces to a couple of scalar problems on subdomains of pure phases.
Here we show that given a solution to the limiting problem under some
additional non degeneracy assumptions there exists a family of solutions
parametrized by the parameter $\beta\gg 1$.
