It is well known that one can specify a complex structure on a real $C^\infty$ manifold in two equivalent ways: an atlas with holomorphic transition functions between charts and an integrable almost complex structure. One of the directions is straightforward. In the other direction, the celebrated Newlander-Nierenberg theorem states that an integrable almost complex structure induces a holomorphic atlas.

For real analytic manifolds, I know only the atlas method (transition functions between charts are real analytic). My question: does there exist for real analytic manifolds an analog of an almost complex structure, an integrability condition and an analog of the Newlander-Nierenberg theorem?

At the moment, my suspicion is that the right analog should be an almost CR structure and a corresponding integrability condition (whatever those are). But unfortunately I've not really seen this written anywhere.