We simulate the viscous evolution of an accretion disc around a spinning black hole. In general, any such disc is misaligned, and warped by the Lense–Thirring effect. Unlike previous studies, we use effective viscosities constrained to be consistent with the internal fluid dynamics of the disc. We find that non-linear fluid effects, which reduce the effective viscosities in warped regions, can promote breaking of the disc into two distinct planes. This occurs when the Shakura & Sunyaev dimensionless viscosity parameter α is ≲0.3 and the initial angle of misalignment between the disc and hole is ≳45°. The break can be a long-lived feature, propagating outwards in the disc on the usual alignment time-scale, after which the disc is fully co-aligned or counter-aligned with the hole. Such a break in the disc may be significant in systems where we know the inclination of the outer accretion disc to the line of sight, such as some X-ray binaries: the inner disc, and so any jets, may be noticeably misaligned with respect to the orbital plane.