We analyse the transverse instability of two-component vector solitons
in a saturable nonlinear medium, in relation to recent experimental
observations of spatial vector soliton pairs in photorefractive media. We present the stability analysis for all three realizations:
bright-bright, dark-dark, and dark-bright solitons, and demonstrate that
both the nonlinearity saturation and incoherent mode interaction can lead
to a strong suppression of the soliton instability.