SUNRISE III is a balloon-borne solar observatory dedicated to the investigation of the processes governing the physics of the magnetic field and the plasma flows in the lower solar atmosphere. The gondola hosts a 1-m aperture telescope that feeds three post-focus instruments. One of these instruments, the Tunable Magnetograph (TuMag), is a tunable imaging spectropolarimeter in visible wavelengths. It is designed to probe the vector magnetic field, $\bf{B}$, and the line-of-sight (LoS) velocity, vLoS, of the photosphere and the lower chromosphere. It provides polarized images with a 63″×63″ field of view (FoV) of the Sun in four polarization states. These images can later be processed on ground to retrieve maps of the aforementioned solar physical quantities. The quasi-simultaneous observation of two spectral lines sensitive to $\bf{B}$ and vLoS in the photosphere and lower chromosphere provides excellent diagnostic measurements of the magnetic and dynamic coupling in these layers. When combined with the other two instruments on board, observing in the infrared and ultraviolet regions of the spectrum, TuMag's diagnostic potential is expected to be greatly enhanced. Building upon heritage of instruments like IMaX and SO/PHI, the key technologies employed for TuMag are a liquid-crystal-variable-retarder-based polarimeter and a solid, LiNbO3 Fabry–Pérot etalon as a spectrometer. However, it also incorporates several innovative features, such as in-house-made, high-sensitivity scientific cameras and a double filter wheel. The latter makes TuMag the first balloon-borne instrument of its type capable of tuning between spectral lines. Specifically, it can sequentially observe any two out of the three spectral lines of Fe I at 525.02 and 525.06 nm and of Mg I at 517.3 nm. Time cadences range from 30 to 100 seconds, depending on the observing mode and the specific pair of spectral lines targeted. Laboratory measurements have demonstrated good image quality, spectral resolution, and polarimetric efficiency. Here we report on the concept, design, calibration, and integration phases of the instrument, as well as on the data reduction pipeline.