jats:pThe interface of a strain-rate-softening fluid that displaces a low-viscosity fluid in a circular geometry with negligible drag can develop finger-like patterns separated by regions in which the fluid appears to be torn apart. Such patterns were observed and explored experimentally in Part 1 using polymeric solutions. They do not occur when the viscosity of the displacing fluid is constant, or when the displacing fluid has no-slip conditions along its boundaries. We investigate theoretically the formation of tongues at the interface of an axisymmetric initial state. We show that finger-like patterns can emerge when circular interfaces of strain-rate-softening fluids displace low-viscosity fluids between stress-free boundaries. The instability, which is fundamentally different from the classical Saffman–Taylor viscous fingering, is driven by the tension that builds up along the circular front of the propagating fluid. That destabilising tension is a geometrical consequence and is present independently of the nonlinear properties of the fluid. Shear stresses stabilise the growth either along extended circumferential streamlines or through a street of vortices. However, such stabilising processes become weaker, thereby allowing the instability to develop, the more strain-rate-softening the fluid is. The theoretical model that we present predicts the main experimental observations made in Part 1. In particular, the patterns we predict using linear-stability theory are consistent with the strongly nonlinear experimental patterns. Our model depends on a single dimensionless number representing the power-law exponent, which implies that the instability we describe could arise in any extensional flow of strain-rate-softening material, ranging from suspensions that rupture in squeeze experiments to rifts formed in ice shelves.</jats:p>