Novel minimally invasive surgical paradigms accessing deep surgical sites present a new challenge of safe instrument insertion and navigation. This paper addresses this challenge by presenting a new framework for compliant motion control of multi-backbone continuum robots subject to whole-arm contacts. This control framework does not rely on knowledge of contact locations along the length of a continuum robot. Instead, the forces at joint level are applied as controller inputs to generate compliant motion. The paper first presents a new mapping of the external wrenches to a generalized force in the configuration space of a single-stage multi-backbone continuum robot. A closed-form analytic expression for the passive stiffness of a multi-backbone continuum robot segment is also presented. A controller, robust to uncertainties of the system model, is proposed to provide compliant motion of the continuum robot segment by using the generalized force and stiffness definitions. Stability, convergence, and controller properties are shown through experimental validation. The presented framework defines a method for providing compliant motion to continuum robots without explicit knowledge of the environment. We believe this work enables new control algorithms for rapidly deployable surgical robots and supports novel surgical paradigms by increasing safety during unstructured interaction with flexible anatomy.