  We prove the existence of an open set of $n_1\times n_2 \times n_3$ tensors
of rank $r$ on which a popular and efficient class of algorithms for computing
tensor rank decompositions based on a reduction to a linear matrix pencil,
typically followed by a generalized eigendecomposition, is arbitrarily
numerically forward unstable. Our analysis shows that this problem is caused by
the fact that the condition number of the tensor rank decomposition can be much
larger for $n_1 \times n_2 \times 2$ tensors than for the $n_1\times n_2 \times
n_3$ input tensor. Moreover, we present a lower bound for the limiting
distribution of the condition number of random tensor rank decompositions of
third-order tensors. The numerical experiments illustrate that for random
tensor rank decompositions one should anticipate a loss of precision of a few
digits.
