  We experimentally investigate second harmonic generation from strongly
coupled localized and propagative phonon polariton modes in arrays of silicon
carbide nanopillars. Our results clearly demonstrate the hybrid nature of the
system's eigenmodes and distinct manifestation of strong coupling in the linear
and nonlinear response. While in linear reflectivity the intensity of the two
strongly-coupled branches is essentially symmetric and well explained by their
respective localized or propagative components, the second harmonic signal
presents a strong asymmetry. Analyzing it in detail, we reveal the importance
of interference effects between the nonlinear polarization terms originating in
the bulk and in the phonon polariton modes, respectively.
