  Let $T$ be a real tensor of (real) rank $r$. $T$ is 'identifiable' when it
has a unique decomposition in terms of rank $1$ tensors. There are cases in
which the identifiability fails over the complex field, for general tensors of
rank $r$. This behavior is quite peculiar when the rank $r$ is submaximal.
Often, the failure is due to the existence of an elliptic normal curve through
general points of the corresponding Segre, Veronese or Grassmann variety. We
prove the existence of nonempty euclidean open subsets of some variety of
tensors of rank $r$, whose elements have several decompositions over $\mathbb
C$, but only one of them is formed by real summands. Thus, in the open sets,
tensors are not identifiable over $\mathbb C$, but are identifiable over
$\mathbb R$.
  We also provide examples of non trivial euclidean open subsets in a whole
space of symmetric tensors (of degree $7$ and $8$ in three variables) and of
almost unbalanced tensors Segre Product ($\mathbb P^2\times \mathbb P^4\times
\mathbb P^9$) whose elements have typical real rank equal to the complex rank,
and are identifiable over $\mathbb R$, but not over $\mathbb C$. On the
contrary, we provide examples of tensors of given real rank, for which real
identifiability cannot hold in non-trivial open subsets.
