Yes, there are generalizations of Rainich conditions to many others fields, including scalar fields, perfect fluids and *null* electromagnetic fields. (Indeed! The original form of the Rainich conditions were not conveniently formulated for electromagnetic fields \(F\in \bigwedge ^2M\) whose both Poincaré invariant vanishes: \(F \wedge \star F = 0\) and \(F \wedge F = 0\). This even lead some very good relativists even to doubt that null-electromagnetic fields could be present in electrovacuum solutions of Einstein-Maxwell equations. Cf. for instance this paper by Louis Witten: "Geometry of Gravitation and Electromagnetism" here. Then Peres and Bonnor found some plane-fronted wave solutions to Einstein-Maxwell and showed that they were perfectly consistent.)

A complete review for all these types of Rainich conditions can be found is:

https://arxiv.org/abs/1308.2323

https://arxiv.org/abs/1503.06311

PS: Peres and Bonnor solutions (which describes some interesting coupled system of electromagnetic-gravitational waves) are here:

http://journals.aps.org/pr/abstract/10.1103/PhysRev.118.1105

http://projecteuclid.org/euclid.cmp/1103841572 (free access)