The most common form of measurement in quantum mechanics projects a wavefunction onto orthogonal states that correspond to definite outcomes. However, generalized quantum measurements that do not fully project quantum states are possible and have an important role in quantum information tasks. Unfortunately, it is difficult to certify that an experiment harvests the advantages made possible by generalized measurements, especially beyond the simplest two-level qubit system. Here we show that multiport beamsplitters allow for the robust realization of high-quality generalized measurements in higher-dimensional systems with more than two levels. Using multicore optical fibre technology, we implement a seven-outcome generalized measurement in a four-dimensional Hilbert space with a fidelity of 99.7%. We present a practical 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 non-projective quantum measurements beyond qubit systems.