An ultrasonic method providing for an efficient global detection of defects in complex media (multiple scattering or reverberating media) is reported herein; this method is based on the nonlinear acoustic mixing of coda waves (stemming from multiple scattering) with lower frequency-swept pump waves. Such a nonlinear mixing step is made possible by the presence of nonlinear scatterers, such as cracks and delamination, yet remains absent when the waves are scattered only by linear scatterers, as is the case in a complex but defect-free medium. A global inspection is achieved thanks to the use of wide-band coda and pump signals, which ensure the excitation of many resonances along with a homogeneous acoustic energy distribution in the medium. We introduce the existing sensitivity tools developed for Coda Wave Interferometry in extracting the pump amplitude-dependent parameters of the coda waves associated with effective nonlinear parameters of the medium. By comparing results at two damage levels, these effective nonlinear parameters are shown to be correlated with crack presence in glass samples. The mechanisms potentially responsible for the observed amplitude dependence on the tested elastic parameters and waveform modification are discussed.