I am new to this area and I am a bit confused by the literature. Is there a strong maximum principle for CR functions over domains in a CR manifold, please? If so, could someone please state it (together with its hypotheses etc.) and provide a reference to a proof (or maybe just prove it if it is not too long)? In the mean time, I will keep on digging in the literature, but in any case, it may be of interest to others too.

Edit 1: it looks like the answer is no, according to ex. 3.2.5 in CR functions (LibreText Mathematics), by Jiří Lebl. Actually, this makes sense, because for example, in the simplest case where our CR manifold is a real hypersurface in $\mathbb{C}^{n+1}$ given by $z_{n+1} - \bar{z}_{n+1} = 0$, then being a CR function is, if I understand the definition well, just being holomorphic in the $z_i$, for $i = 1, \ldots, n$, but it says nothing about how the function depends on $x_{n+1}$ ($x_{n+1}$ is the real part of $z_{n+1}$).

But in my case, the function is also real analytic. So allow me to ask if there is a strong maximum principle for *real analytic* CR functions on a domain inside a CR manifold. I'd also be happy to assume that (if it helps), at each point $p \in M$, the $T_p^{(1,0)}$ bundle of $M$ has complex rank $n$, where the dimension of $M$ is $2n+1$ (so that, in some sense, $M$ is like a real hypersurface inside some complex $n+1$ dimensional manifold).