Continuum manipulator compliance enables operation in delicate environments while challenging the actuation and control approaches. In the case of a catheter ablation of atrial fibrillation, the compliance of the continuum backbone lends an inherent safety to the device. This inherent safety frustrates attempts at precise, accurate, and fast control, limiting the continuum devices to simple and static positioning tasks. This paper develops interleaved continuum-rigid manipulation, by which the hysteretic nonlinearities encountered in tendon-actuated continuum manipulators are compensated by the discrete rigid joints located between the continuum sections. The rigid joints introduce an actuation redundancy, which an interleaved controller may use to avoid the continuum nonlinearities and dynamic excitations, or to prefer particular configurations that may improve task accuracy, permit greater end-effector forces, or avoid environment obstacles. Two experimental systems explore the potential of these joints to: 1) increase the manipulator's dexterous workspace, and 2) correct for actuation nonlinearities and enhance manipulator performance. These experiments also expose important design and control observations that were not apparent in the general robotic and continuum literature.