  Inspired by a result of Wong (Commun. Partial Differ. Equ. 13(10):1209-1221,
1988), we establish an analytic description of the essential spectrum of
non-self-adjoint mixed-order systems of pseudo-differential operators on
$L^2(\mathbb{R}^N) \oplus L^2(\mathbb{R}^N)$ that are uniformly
Douglis-Nirenberg elliptic with positive-order diagonal entries. We apply our
result to a problem arising in the dynamics of falling liquid films.
