Let me begin with a few definitions. My question will be basically how to simplify them to something more manageable. The motivation for these definitions is given at the end.
Let $X$ be a topological space (or even a locale). If $U$ is an open set in $X$, I will be writing $\newcommand{\negation}{\mathop{\sim}} \negation U$ for the largest open set disjoint with $U$.
Recall that an open set $U$ is said to be regular when $(\negation\negation U) = U$, or equivalently, when $U$ is of the form $\negation V$ for some open $V$.
Let us say say that an open set is sewn¹ when it is of the form $U\cup(\negation U)$ for some regular open set $U$, or, equivalently, $(\negation V)\cup(\negation\negation V)$ for some open $V$. Clearly, sewn open sets are dense; see Example of a dense open subset that is not of the form $U \cup \mathop\sim U$ for $U$ regular on MSE (and J. D. Hamkins's answer to it) for a partial converse.
It is worth noting that the intersection of two (hence, finitely many) sewn sets is sewn: indeed, if $U_1$, $U_2$ are regular, then $(U_1\cup(\negation U_1)) \cap (U_2\cup(\negation U_2)) = (U\cup(\negation U))$ where $U = (U_1\cap U_2) \cup ((\negation U_1)\cap(\negation U_2))$ and $\negation U = ((\negation U_1)\cap U_2) \cup (U_1\cap(\negation U_2))$ are regular, as is not difficult to check. [Edit: As pointed out in a comment, this is wrong. The following definition has been altered accordingly.]
Let us say that an open set $D$ is DM-dense when it contains a finite intersection² of sewn sets, i.e., when there are $U_1,\ldots,U_n$ regular open such that $D \supseteq \bigcap_{i=1}^n(U_i\cup(\negation U_i))$). This defines a filter of open sets, and clearly every DM-dense open set is dense.
Finally, let us say that an open set $W$ is DeMorganian when every open set $W'$ such that $W \subseteq W'$ and such that $W \cap D = W' \cap D$ for some DM-dense open set $D$ is, in fact, equal to $W$. In other words, $W$ is the greatest in its equivalence class for the equivalence relation given by “$W \cap D = W' \cap D$ for some DM-dense $D$” (this is indeed an equivalence relation since the DM-dense sets are a filter).
I will be saying in the motivation below why these sets are interesting, and also why regular open sets are DeMorganian, but first, here is my question:
Question: Can we describe DeMorganian open sets in a more simple manner, hopefully avoiding the “sewn” terminology altogether? Or, if $W$ is an open set, can we describe its DeMorganization $j_{\mathrm{DM}}(W)$, namely the largest open set $W'$ such that $W \cap D = W' \cap D$ for some DM-dense $D$?
Motivation: These definitions are adapted from the paper “De Morgan classifying toposes” by Olivia Caramello, theorem 1.10: the latter describes the largest dense De Morgan sublocale of $X$, where a “De Morgan locale” is one whose frame of open sets satisfies the weak law of excluded middle ($(\negation V) \cup (\negation\negation V)$ is full for any $V$), or equivalently, the De Morgan laws (hence the name). What I'm hoping for is something analogous to the description of Booleanization:
Analogy: An open set $W$ such that “every open set $W'$ such that $W \subseteq W'$ and such that $W \cap D = W' \cap D$ for some dense set $D$ is, in fact, equal to $W$” is precisely the same as a regular open set, and the largest such $W'$, call it $j_{\neg\neg}(W)$, is simply $\negation\negation W$, the regularization of $W$. The set of regular open sets is the largest dense Boolean sublocale of $X$, where a “Boolean locale” is one whose frame of open sets is a Boolean algebra.
Note: It follows from the previous paragraph that regular open sets are DeMorganian (we have $j_{\mathrm{DM}}(W) \leq j_{\neg\neg}(W)$). Of course, when every dense set is DM-dense, and in particular if every dense set is sewn, as is the case of $\mathbb{R}$ by J. D. Hamkins's response mentioned above, the DeMorganian open sets are just the regular open sets and the question is vacuous.
At the other extreme, if $X$ is extremally disconnected (Hausdorff?), then its frame of opens is De Morgan (the only sewn open set is $X$ itself, so it's also the only DM-dense set) and every open set is DeMorganian (but it is generally not the case that every open set is regular, thus providing examples of DeMorganian open sets that are not regular).
For lack of a better word, I had to come up with something. (Imagine stitching $U$ and $\negation U$ together in a failed attempt to cover $X$. Note that the two parts play a completely symmetric role.)
I don't know if this finite intersection is really needed. (A previous version of this question claimed incorrectly that a finite intersection of sewn sets is sewn, which is wrong as was pointed out by J. D. Hamkins in the comments, but this doesn't rule out that a finite intersection of sewn sets contains a sewn set, which would simplify the definition and make sewn sets a filter basis for the DM-dense sets.) Note that O. Caramello's paper (arXiv version) has $a \geq \bigvee_{1\leq i\leq n}(u_i\vee\neg u_i)$ on page 9 (line −5), which I believe should be $a \geq \bigwedge_{1\leq i\leq n}(u_i\vee\neg u_i)$.
