Thanks to a question I previously asked, I realized that a Gaussian integer matrix should have a determinant of $\pm 1$ or $\pm i$ for it to have an Gaussian integer inverse. From that, I gather that if one considers an Eisenstein integer matrix, it should have a determinant of $\pm 1$, $\pm \exp(\pm 2\pi i/3)$. For the Gaussian integer case at least, I can generate random Gaussian integer matrices through the reduction to Hermite normal form.
I now ask if there is an algorithm for generating a random Eisenstein integer matrix whose inverse has Eisenstein integer entries.