  We introduce and investigate a class of complex semi-infinite banded Toeplitz
matrices satisfying the condition that the spectra of their principal
submatrices accumulate onto a real interval when the size of the submatrix
grows to $\infty$. We prove that a banded Toeplitz matrix belongs to this class
if and only if its symbol has real values on a Jordan curve located in
$\mathbb{C}\setminus\{0\}$. Surprisingly, it turns out that, if such a Jordan
curve is present, the spectra of all the submatrices have to be real. The
latter claim is also proven for matrices given by a more general symbol.
Further, the limiting eigenvalue distribution of a real banded Toeplitz matrix
is related to the solution of a determinate Hamburger moment problem. We use
this to derive a formula for the limiting measure using a parametrization of
the Jordan curve. We also describe a Jacobi operator, whose spectral measure
coincides with the limiting measure. We show that this Jacobi operator is a
compact perturbation of a tridiagonal Toeplitz matrix. Our main results are
illustrated by several concrete examples; some of them allow an explicit
analytic treatment, while some are only treated numerically.
  Update: The proof of Theorem 8 contains an error. An erratum is attached in
the end
