The bilateral simultaneous generation of sound in some oscine songbirds leads to complex sounds that cannot be described in terms of a superposition of the isolated sources alone. In this work, we study the appearance of complex solutions in a model for the acoustic interaction between the two sound sources in birdsong. The origin of these complex oscillations can be traced to the nonlinear mode-mode interaction arising when both sources are active. As an example, we analyze a remarkable sound produced by an oscine songbird and show that the proposed dynamical scenario is compatible with the observed behavior.