Rembrandt lighting is a well-known technique to produce Chiaroscuro, presentation of depth using light and shadow areas.
But in addition to Chiaroscuro, there were three other painting modes in the Renaissance - Sfumato, Cangiante and Unione.
Are those styles also applicable in photography, and if yes, how would one reproduce them?
Purely photographic implementations of cangiante (literally "changing", but figuratively "tone forcing") are actually quite common, using warmer light in the key (main light) and cooler in the fill. Every time you use a gold reflector to light a subject against an inky blue sky at twilight, you are using cangiante. The actual greyscale tonal value difference between highlights and shadows is much smaller than with chiaroscuro; the difference in colour temperature between the two gives the same sort of impression of depth with a much smaller value variation. The Impressionist painting (and photographic) style is similar, playing warm highlights against cool shadows to give the impression of greater tonal variation. This allows images to be relatively high in key while maintaining "body" in the subject shapes.
Sfumato can be approximated with nearly-direct (axial) soft light (either a large softbox or a ringlight), often combined with soft focus or a diffusion/star filter. Getting to the point of actually losing all sharpness of detail would probably be unacceptable for "straight" photography due to the viewer's expectations of the medium, though it might work well in a fantasy fine art mode.
Pulling back from the edge of blurriness and giving some substance to edges without emphasizing them (as one would in chiaroscuro), you arrive at unione. Again, the tonal key tends to be relatively high, or at least the darks are moved up the tone scale somewhat as compared to chiaroscuro. (Originally, that was more of a limitation of the medium than anything else -- it is hard to get real darks in egg tempera or fresco other than black, and easy in oils.) Soft light, not too far away from axial (call forty-five degrees either side of the camera an absolute limit unless the source is large enough to provide substantial wrapping at larger angles) combined with a relatively high degree of fill ought to get you close to the Renaissance ideal. The exact placement and depth of shadow is a matter of personal preference, but if anything falls off to total darkness, or if the shadows are too well defined, you are in chiaroscuro territory.
