  Many experiments with high-dimensional quantum systems aim at observing
physical properties without having to assume that all measurement devices are
perfectly characterised. To implement high-dimensional measurements, it is a
common practice to separately project the system onto each of the possible
eigenstates followed by post-processing. Here, we argue that such measurement
binarisation procedures are flawed and open a loophole in black-box correlation
experiments. We point out how binarised measurements can lead to meaningless
results unless artificial assumptions are introduced. We then propose a method
to faithfully analyse correlations from binarised measurements. We apply this
approach to well-known examples of quantum correlations in Bell,
prepare-and-measure and steering experiments. For the former two, we find that
binarisation permits reduced-but-yet-significant quantum advantage, while for
the latter we find that binarisation can sometimes lead to undiminished quantum
correlations. Our discussion emphasises the importance of both more careful
data analysis and for developing genuine multi-outcome quantum measurements.
