Below, "structure" means "computable structure in a computable language." In particular, we do distinguish between isomorphic copies of the same structure.
Let $\mathcal{L}_{\omega_1,\omega}$ be defined so that $(i)$ the only Boolean operations allowed are
$\bigwedge_{i<n}$ and $\bigvee_{i<n}$ for $n\le\omega$, as well as
$\rightarrow$ and $\neg$, and
$(ii)$ negations are only allowed at atomic formulas. Given a computable structure $\mathfrak{A}$ with domain $\omega$, we define the relation "$e$ realizes $\varphi$ in $\mathfrak{A}$" - in symbols, "$\mathfrak{A}\models^e\varphi$" - for $\varphi$ a not-necessarily-computable $\mathcal{L}_{\omega_1,\omega}$-sentence with natural number parameters recursively as follows:
If $\varphi$ is atomic or negated atomic, then $\mathfrak{A}\models^e\varphi$ iff $\mathfrak{A}\models\varphi$ in the usual sense.
If $\varphi$ has the form $\psi\rightarrow\theta$, then $\mathfrak{A}\models^e\varphi$ iff $e$ is an index for a partial computable functional $\Phi$ such that, whenever $\mathfrak{A}\models^c\psi$, we have $\mathfrak{A}\models^{\Phi^{\varphi}(c)}\theta$. (Since $\varphi$ is itself a countable object, essentially a labelled subtree of $\omega^{<\omega}$, it makes sense for $\Phi$ to take $\varphi$ as an oracle.)
If $\varphi$ has the form $\bigvee_{i<n}\psi_i$, then $\mathfrak{A}\models^e\varphi$ iff $e=\langle c,d\rangle$ such that $d<n$ and $\mathfrak{A}\models^c\psi_d$.
If $\varphi$ has the form $\bigwedge_{i<n}\psi_i$, then $\mathfrak{A}\models^e\varphi$ iff $e$ is an index for a partial computable functional $\Phi$ such that for each $i<n$ we have $\mathfrak{A}\models^{\Phi^\varphi(i)}\psi_i$.
If $\varphi$ has the form $\forall x\psi$, then $\mathfrak{A}\models^e\varphi$ iff $e$ is an index for a total computable function $p$ such that for all $n\in\omega$, $\mathfrak{A}\models^{p(n)}\psi[n/x]$ (where "$\psi[n/x]$" is the result of replacing each free occurrence of $x$ in $\psi$ with $n$).
If $\varphi$ has the form $\exists x\psi$, then $\mathfrak{A}\models^e\varphi$ iff $e=\langle c,d\rangle$ such that $\mathfrak{A}\models^c\psi[d/x]$.
Write "$\mathfrak{A}\models^\star\varphi$" if $\mathfrak{A}\models^e\varphi$ for some $e\in\omega$. Finally, for $\mathfrak{A}$ a computable structure, say that a sentence $\varphi$ effectively captures $\mathfrak{A}$ iff for all computable structures $\mathfrak{B}$, the following are equivalent:
$\mathfrak{B}\models^\star\varphi$.
$\mathfrak{A}$ is computably isomorphic to $\mathfrak{B}$.
It's not hard to show (via Barwise compactness) that if we require $\varphi$ to be computable, at least some Harrison orders are not effectively capturable. However, this argument breaks down when the sentence $\varphi$ is allowed to be of high complexity.
Question: Is every computable structure effectively capturable (by not-necessarily-computable sentences)?
It's not hard to show that any counterexample must have computable dimension (= number of isomorphic computable copies up to computable isomorphism) $>1$, but beyond that I don't see anything useful. It's not even clear to me that any counterexample must have infinite computable dimension.
