We explore the relationship between the hard X-ray and submillimetre populations using deep Chandra observations of a large, contiguous SCUBA survey. In agreement with other recent findings, we confirm that the direct overlap is small. Of the 17 submillimetre sources detected in this field at 850 μm, only one is coincident with a Chandra source. The resulting limits imply that the majority of SCUBA sources are not powered by acive galactic nuclei (AGN), unless the central engine is obscured by Compton-thick material with a low (<1 per cent) scattered component. Furthermore, since Chandra detects only ~5 per cent of SCUBA sources, the typical obscuration would need to be almost isotropic. The X-ray upper limits are so strong that in most cases we can also rule out a starburst spectral energy distribution at low redshift, suggesting that the majority of SCUBA sources lie at z > 1, even if they are purely starburst galaxies. Despite the low detection rate, we find evidence for strong angular clustering between the X-ray and submillimetre populations. The implication is that AGN and SCUBA sources trace the same large-scale structure but do not generally coincide. If bright submillimetre sources represent massive elliptical galaxies in formation, we suggest that (for a given galaxy) the major episode of star formation must be distinct from the period of observable quasar activity.