The behaviour in simple shear of two concentrated and strongly cohesive mineral suspensions showing highly non-monotonic flow curves is described. Two rheometric test modes were employed, controlled stress and controlled shear-rate. In controlled stress mode the materials showed runaway flow above a yield stress, which, for one of the suspensions, varied substantially in value and seemingly at random from one run to the next, such that the up flow-curve appeared to be quite irreproducible. The down-curve was not though, as neither was the curve obtained in controlled rate mode, which turned out to be triple-valued in the region where runaway flow was seen in controlled rising stress. For this first suspension, the total stress could be decomposed into three parts to a good approximation: a viscous component proportional to a plastic viscosity, a constant isostatic contribution, and a third shear-rate dependent contribution associated with the particulate network which decreased with increasing shear-rate raised to the -7/10th power. In the case of the second suspension, the stress could be decomposed along similar lines, although the strain-rate softening of the solid-phase stress was found to be logarithmic and the irreducible isostatic stress was small. The flow curves are discussed in the light of recent simulations and they conform to a very simple but general rule for non-monotonic behaviour in cohesive suspensions and emulsions, namely that it is caused by strain-rate softening of the solid phase stress. (C) 2014 Elsevier B.V. All rights reserved.