We identify phase noise as a bottleneck for the performance of digital self-interference cancellers that utilize a single auxiliary receiver---single-tap digital cancellers---and operate in multipath propagation environments. Our analysis demonstrates that the degradation due to phase noise is caused by a mismatch between the analog delay of the auxiliary receiver and the different delays of the multipath components of the self-interference signal. We propose a novel multi-tap digital self-interference canceller architecture that is based on multiple auxiliary receivers and a customized Normalized-Least-Mean-Squared (NLMS) filtering for self-interference regeneration. Our simulation results demonstrate that our proposed architecture is more robust to phase noise impairments and can in some cases achieve 10~dB larger self-interference cancellation than the single-tap architecture.