The nonlinear dynamics exhibited by a planar layer of precessing fluid is examined as a canonical example of a strained rotating flow. The simple basic flow, Ubasic = −YXˆ+(X−2εZ)Yˆ in a frame rotating at εXˆ, consists of sheared circular streamlines (where ε measures the shearing) which are linearly unstable through the pairwise resonance of two inertial waves in a fashion similar to elliptical flow. Direct numerical simulation shows that the weakly nonlinear regime is quickly disrupted by further instabilities which lead to a multitude of co-existing solution branches, some of which represent chaotic flows. All these solutions remain within O(ε) (in an energy norm) of Ubasic so that energy is not apparently withdrawn from the fluid's underlying rotation. Further increases in the precession rate cause the flow to branch-switch randomly between these now quasi-stable states so that a new form of ‘slow’ dynamics emerges. The implication of this and the fact that these instabilities can nevertheless be classed as ‘strong’ is discussed from the perspective of the closely related problem of the precessing Earth and laboratory models thereof.