  To investigate the imprint on the gravitational-wave emission from extreme
mass-ratio inspirals in non-pure Kerr spacetimes, we have studied the
``kludge'' waveforms generated in highly-accurate, numerically-generated
spacetimes containing a black hole and a self-gravitating, homogeneous torus
with comparable mass and spin. In order to maximize their impact on the
produced waveforms, we have considered tori that are compact, massive and close
to the central black hole, investigating under what conditions the LISA
experiment could detect their presence. Our results show that for a large
portion of the space of parameters the waveforms produced by EMRIs in these
black hole-torus systems are indistinguishable from pure-Kerr waveforms. Hence,
a ``confusion problem'' will be present for observations carried out over a
timescale below or comparable to the dephasing time.
