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Nonlinear interactions are crucial in science and engineering. Here, weinvestigate wave interactions in a highly nonlinear magnetic system driven byparametric pumping leading to Bose--Einstein condensation of spin-wave quanta-- magnons. Using Brillouin light scattering spectroscopy in yttrium-irongarnet films, we found and identified a set of nonlinear processes resulting inoff-resonant spin-wave excitations -- virtual magnons. In particular, wediscovered a dynamically-strong, correlation-enhanced four-wave interactionprocess of the magnon condensate with pairs of parametric magnons havingopposite wavevectors and fully correlated phases.

Correlation-enhanced interaction of a Bose-Einstein condensate with parametric magnon pairs and virtual magnons