  Generalised quantum measurements go beyond the textbook concept of a
projection onto an orthonormal basis in Hilbert space. They are not only of
fundamental relevance but have also an important role in quantum information
tasks. However, it is highly demanding to certify that a generalised
measurement is indeed required to explain the results of a quantum experiment
in which only the degrees of freedom are assumed to be known. Here, we use
state-of-the-art multicore optical fiber technology to build multiport
beamsplitters and faithfully implement a seven-outcome generalised measurement
in a four-dimensional Hilbert space with a fidelity of $99.7\%$. We apply it to
perform an elementary quantum communication task and demonstrate a success rate
that cannot be simulated in any conceivable quantum protocol based on standard
projective measurements on quantum messages of the same dimension. Our
approach, which is compatible with modern photonic platforms, showcases an
avenue for faithful and high-quality implementation of genuinely nonprojective
quantum measurements beyond qubit systems.
