Given G a complex connected semisimple Lie group, G0 a connected real form of it and X a smooth complex projective curve, one can define the notion of a G0-Higgs bundle. These objects generalise the usual notion, and also appear naturally in some constructions such as the Hitchin-Kostant section related to the Hitchin sytem of G-Higgs bundles. The Hitchin system for G-Higgs bundles on X is a rich source of geometric information. Donagi and Gaitsgory proved the system for the regular stack of G-Higgs bundles to be a trivial banded gerbe with abelian structure sheaf. Furthermore, they identify the fibers as categories of torsors over Donagi's cameral cover.

In this talk I will present work in progress on the Hitchin system for G0-Higgs bundles. In particular, I will explain a cameral construction in this context, focusing on the unramified case. The main difference with the complex case is that fibers need not be abelian (although they are for example in the case of split real forms). I will give some examples illustrating this phenomenon.