It was recently experimentally proved that the superconducting counterpart of a diode, i.e., a device that realizes nonreciprocal Cooper pairs transport, can be realized by breaking the spatial and time-reversal symmetry of a system simultaneously. Here, we report the theory, fabrication, and operation of a monolithic dc superconducting quantum interference device that embedding three-dimensional Dayem nanobridges as weak links realizes an efficient and magnetic flux-tunable supercurrent diode. The device is entirely realized in Al and achieves a maximum rectification efficiency of ∼20%, which stems from the high harmonic content of its current-to-phase relation only without the need of any sizable screening current caused by a finite loop inductance. Our interferometer can be easily integrated with state-of-the-art superconducting electronics and since it does not require a finite loop inductance to provide large rectification, its downsizing is not limited by the geometrical constraints of the superconducting ring.