  We compute the signed distribution of the eigenvalues/vectors of the complex
order-three random tensor by computing a partition function of a four-fermi
theory, where signs are from a Hessian determinant associated to each
eigenvector. The issue of the presence of a continuous degeneracy of the
eigenvectors is properly treated by a gauge-fixing. The final expression is
compactly represented by a generating function, which has an expansion whose
powers are the dimensions of the tensor index spaces. A crosscheck is performed
by Monte Carlo simulations. By taking the large-$N$ limit we obtain a critical
point where the behavior of the signed distribution qualitatively changes, and
also the end of the signed distribution. The expected agreement of the end of
the signed distribution with that of the genuine distribution provides a few
applications, such as the largest eigenvalue, the geometric measure of
entanglement, and the best rank-one approximation in the large-$N$ limit.
