ESO's two FOcal Reducer and low-dispersion Spectrographs (FORS) are the primary optical imaging instruments for the VLT. They are not direct-imaging instruments, as there are several optical elements in the light path. In particular, both instruments are attached to a field rotator. Obtaining truly photometric data with such instruments presents a significant challenge. In this article, we investigate in detail twilight flats taken with the FORS instruments. We find that a large fraction of the structure seen in these flat fields rotates with the field rotator. We discuss in detail the methods we use to determine the cause of this effect. The effect was tracked down to be caused by the Linear Atmospheric Dispersion Corrector (LADC). The results are thus of special interest for designers of instruments with LADCs and developers of calibration plans and pipelines for such instruments. The methods described here to find and correct it, however, are of interest also for other instruments using a field rotator. If not properly corrected, this structure in the flat field may degrade the photometric accuracy of imaging observations taken with the FORS instruments by adding a systematic error of up to 4% for broadband filters. We discuss several strategies to obtain photometric images in the presence of rotating flat-field pattern.