Compressive sensing (CS) combines data acquisition with compression coding to reduce the number of measurements required to reconstruct a sparse signal. In optics, this usually takes the form of projecting the field onto sequences of random spatial patterns that are selected from an appropriate random ensemble. We show here that CS can be exploited in 'native' optics hardware without introducing added components. Specifically, we show that random sub-Nyquist sampling of an interferogram suffices to reconstruct the field modal structure despite the structural constraints of the measurement system set by its limited degrees of freedom. The distribution of the reduced (and structurally constrained) sensing matrices corresponding to random measurements is provably incoherent and isotropic, which helps us carry out CS successfully. We implement compressive interferometry using a generalized Mach-Zehnder interferometer in which the traditional temporal delay is replaced with a linear transformation corresponding to a fractional transform. By randomly sampling the order of the fractional transform, we efficiently reconstruct the modal content of the input beam in the Hermite-Gaussian and Laguerre-Gaussian bases.