The late-time effect of primordial non-Gaussianity offers a window into the physics of inflation and the very early Universe. In this work we study the consequences of a particular class of primordial non-Gaussianity that is fully characterized by initial density fluctuations drawn from a non-Gaussian probability density function, rather than by construction of a particular form for the primordial bispectrum. We numerically generate multiple realisations of cosmological structure and use the late-time matter polyspectra to determine the effect of these modified initial conditions. In this non-Gaussianity has only a small imprint on the first polyspectra, when compared to a standard Gaussian cosmology. Furthermore, some of our models present an interesting scale-dependent deviation from the Gaussian case in the bispectrum and trispectrum, although the signal is at most at the percent level.