  The geometry of the set of restrictions of rank-one tensors to some of their
coordinates is studied. This gives insight into the problem of rank-one
completion of partial tensors. Particular emphasis is put on the semialgebraic
nature of the problem, which arises for real tensors with constraints on the
parameters. The algebraic boundary of the completable region is described for
tensors parametrized by probability distributions and where the number of
observed entries equals the number of parameters. If the observations are on
the diagonal of a tensor of format $d\times\dots\times d$, the complete
semialgebraic description of the completable region is found.
