The stability of the phase difference between two white-light continua, generated from the same 180-fs pulses at ≃1035 nm, is assessed by a modified Bellini-Hänsch interferometer. Mutual spectral phase stability is studied and quantified as a function of several parameters: pulse energy, position of the nonlinear crystal with respect to the beam waist and interaction length. Our results show that intrapulse decoherence may significantly contribute to the measured CEP noise floor. In addition, spectrally-resolved intensity-to-phase coupling coefficients are measured and stability regions are identified.