We present what we believe to be a novel method for order sorting a Fabry–Perot interferometer using a Fourier transform spectrometer (FTS) in tandem. We demonstrate how the order sorting is achieved using a model instrument response as an example of an instrument working in the 5–25\text{\hspace{0.17em} μm} band, although the method is generally applicable at all wavelengths. We show that an instrument of this type can be realized with a large bandwidth, a large field of view, and good transmission efficiency. These attributes make this instrument concept a useful technique in applications where true imaging spectroscopy is required, such as mapping large astronomical sources. We compare the performance of the new instrument to grating and standard FTS instruments in circumstances where the measurement is background and detector noise limited. We use a figure of merit based on the field of view and speed of detection and find that the new system has a speed advantage over a FTS with the same field of view in all circumstances. The instrument will be faster than a grating instrument with the same spectral resolution once the field of view is >\text{13} times larger under high background conditions and >\text{50} times larger with detector performances that match the photon noise from Zodiacal light.