  We are concerned with the eigenstructure of supersymmetric tensors. Like in
the matrix case, normalized tensor eigenvectors are fixed points of the tensor
power iteration map. However, unless the given tensor is orthogonally
decomposable, some of these fixed points may be repelling and therefore be
undetectable by any numerical scheme. In this paper, we consider the case of
regular simplex tensors whose symmetric decomposition is induced by an
overcomplete, equiangular set of $n+1$ vectors from $\mathbb R^n$. We discuss
the full real eigenstructure of such tensors, including the robustness analysis
of all normalized eigenvectors. As it turns out, regular simplex tensors
exhibit robust as well as non-robust eigenvectors which, moreover, only partly
coincide with the generators from the symmetric tensor decomposition.
