  We demonstrate an experimental test of the Clauser-Horne-Shimony-Holt (CHSH)
Bell inequality which seemingly exhibits correlations beyond the limits imposed
by quantum mechanics. Inspired by the idea of Fourier synthesis, we design
analysers that measure specific superpositions of orbital angular momentum
(OAM) states, such that when one analyser is rotated with respect to the other,
the resulting coincidence curves are similar to a square-wave. Calculating the
CHSH Bell parameter, $S$, from these curves result to values beyond the
Tsirelson bound of $S_{QM}=2\sqrt{2}$. We obtain $S=3.99\pm0.02$, implying
almost perfect nonlocal Popescu-Rohrlich correlations. The "super-quantum"
values of $S$ is only possible in our experiment because our experiment,
subtly, does not comply with fair-sampling. The way our Bell test fails
fair-sampling is not immediately obvious and requires knowledge of the states
being measured. Our experiment highlights the caution needed in Bell-type
experiments based on measurements within high-dimensional state spaces such as
that of OAM, especially in the advent of device-independent quantum protocols.
