Josephson tunnelling between YBa₂Cu₃O_7-δ and Pb with the current flowing along the c-axis of the YBa₂Cu₃O_7-δ is presumed to arise from an s-wave component of the superconductivity in YBa₂Cu₃O_7-δ. Experiments on multi-twin samples are not entirely consistent with this hypothesis. The sign changes of the s-wave order parameter across the N_T twin boundaries should give cancellations, resulting in a small ((N_T)^1/2) tunnelling current. The actual current is larger than this. We present a theory of this unexpectedly large current based upon a surface effect: disorder-induced suppression of the d-wave component at the (001) surface leads to s-wave coherence across the twin boundaries and a non-random tunnelling current. We solve the case of an ordered array of d + s and d-s twins, and estimate that the twin size at which s-wave surface coherence occurs is consistent with typical sizes observed in experiments. In this picture, there is a phase difference of π/2 between different surfaces of the material. We propose a corner-junction experiment to test this picture.