The Bäcklund-gauge transformation for a system of coupled NLS (nonlinear Schro¨dinger) equations with a degenerate associated spectral operator is derived from an algebraic perspective, extending aspects of other results [M. Boiti, Tu. Guizhang, Il Nuovo Cimento 71B (1982) 253–264; D.H. Sattinger, V.D. Zurkowski, Physica D 26 (1–3) (1987) 225–250] that apply in the context of non-degenerate spectral operators. Moreover, we demonstrate how the Ba¨cklund-gauge transformation can be used to explicitly construct the entire unstable manifold (via superpositions of homoclinic orbits) of a plane wave solution with both self-phase instabilities and coupling instabilities. This work builds on the results of Ercolani et al. [N. Ercolani, M.G. Forest, D.W. McLaughlin, Physica D 18 (1986) 472–474; N. Ercolani, M.G. Forest, D.W. McLaughlin, Physica D 43 (2–3) (1990) 349–384] for the sine-Gordon equation, and Forest et al. [M.G. Forest, D.W. McLaughlin, D.J. Muraki, O.C. Wright, J. Nonlinear Sci., in press; M.G. Forest, S.P. Sheu, O.C. Wright, Phys. Lett. A, in press] for the integrable coupled NLS system.