  We extend the nonconforming Trefftz virtual element method introduced in
arXiv:1805.05634 to the case of the fluid-fluid interface problem, that is, a
Helmholtz problem with piecewise constant wave number. With respect to the
original approach, we address two additional issues: firstly, we define the
coupling of local approximation spaces with piecewise constant wave numbers,
secondly, we enrich such local spaces with special functions capturing the
physical behaviour of the solution to the target problem. As these two issues
are directly related to an increase of the number of degrees of freedom, we use
a reduction strategy inspired by arXiv:1807.11237, which allows to mitigate the
growth of the dimension of the approximation space when considering $h$- and
$p$-refinements. This renders the new method highly competitive in comparison
to other Trefftz and quasi-Trefftz technologies tailored for the Helmholtz
problem with piecewise constant wave number. A wide range of numerical
experiments, including the $p$-version with quasi-uniform meshes and the
$hp$-version with isotropic and anisotropic mesh refinements, is presented.
