  We investigate the relative phase coherence properties and the occurrence of
demixing instabilities for two mutually interacting and time evolving
Bose-Einstein condensates in traps. Our treatment naturally includes the
additional decoherence effect due to fluctuations in the total number of
particles. Analytical results are presented for the breathe-together solution,
an extension of previously known scaling solution to the case of a binary
mixture of condensates. When the three coupling constants describing the
elastic interactions among the atoms in the two states are close to each other,
a dramatic increase of the phase coherence time is predicted. Numerical results
are presented for the parameters of the recent JILA experiments.
