This question basically follows this earlier question of mine but shifting from standard systems of nonstandard models of $PA$ to $\omega$-models of $RCA_0$. For $X$ a Turing ideal we get the map $c_X$ on $2^\omega$ given by $c_X(x)=[b[x]]\cap X$ where $b$ is some computable bijection $\omega\cong 2^{<\omega}$; in the language of the linked question, this is a **closed set pattern**.

I'm trying to understand how much information this construction loses in the sense of second-order arithmetic. For $X$ a Turing ideal we *also* get an $\omega$-model of $RCA_0$ - **which I'll freely conflate with $X$ itself** - and I'm interested in what axioms of second-order arithmetic we can detect topologically.

(Below, $\sim$ denotes "difference by homeomorphism" - for $c,d$ closed set patterns on $\mathcal{X},\mathcal{Y}$ we write $c\sim d$ iff for some $H:\mathcal{X}\cong\mathcal{Y}$ we have $a\in c(b)\leftrightarrow H(a)\in d(H(b))$ for all $a,b\in\mathcal{X}$.)

Say that a theory of second-order arithmetic is *topologically detectable* if there is some $\sim$-respecting property of closed set patterns which holds of $c_X$ iff $X\models A$ for each Turing ideal $X$. For example:

$WKL_0$ is topologically detectable via "$c_X^{-1}(\emptyset)$ is open."

$ACA_0$ is topologically detectable in at least two ways

*(after adding $WKL_0$)*: via "$ran(c_X)$ is closed under (single) Cantor-Bendixson derivatives" and "for each $x\in X$ the set $\{y\in X: c(y)\supseteq c(x)\}$ is closed."

My question is essentially whether any other "weak combinatorial principles" are topologically detectable:

Is there some "reasonably natural" $A$ with $ACA_0\models A$ and $WKL_0\not\models_\omega A$ such that $A$ (or at least $WKL_0+A$) is topologically detectable?

*(Here $\models_\omega$ is the restriction of $\models$ to $\omega$-models, which is needed to rule out e.g. $I\Sigma_{17}$. Note that if $A$ is topologically detectable then so is $WKL_0+A$, but the converse isn't obvious to me.)*

The most tempting candidate is of course $RT^2_2$, but there are plenty of others.