I'm reading Woodin's paper "A Discontinuous Homomorphism from C(X) without CH". In this paper, Woodin defined "$\mathbb{R}$-complete" as the following:
DEFINITION 1. Suppose that $\mathbb{H} \subset \mathbb{F}$ are real-closed fields and $\mathbb{R} \subset \mathbb{F}$. The subfield $\mathbb{H}$ is $\mathbb{R}$-complete if whenever $(A,B)$ is an $(\omega,\omega)$ gap in $\mathbb{G}$, either $(A,B)$ is a gap in $\mathbb{H}$ or $r \in \mathbb{H}$ where $r$ is the real defined by $(A,B)$, that is, $r = \sup A $ in $\mathbb{R}$.
What does "$\mathbb{G}$" mean? He asserts that "$\mathbb{H}$ is $\mathbb{R}$ -complete" is equivalent to "$\mathbb{H} \cap \mathbb{R}$ is equal to the residue field of $\mathbb{H}$". However, this does not refer to $\mathbb{G}$.
It appears that $\mathbb{G}$ is a typo and should be $\mathbb{Q}$ instead. An $(\omega,\omega)$-gap in $\mathbb{Q}$ is then just a gap in $\mathbb{Q}$ where an irrational number is, so $\mathbb{H}$ being $\mathbb{R}$-complete just means that if it contains any element in such a gap, it contains the real number in it. That is, whenever $\mathbb{H}$ contains an element that is infinitesimally close to an irrational number, it contains that irrational number. (Or equivalently, whenever $\mathbb{H}$ contains an element that is infinitesimally close to a real number, it contains that real number.)