$\Sigma$-Definability of countable structures over real numbers, complex numbers, and quaternions

We study $\Sigma$-definability of countable models over hereditarily finite ($\mathbb{HF}$-) superstructures over the field $\mathbb R$ of reals, the field $\mathbb C$ of complex numbers, and over the skew field $\mathbb H$ of quaternions. In particular, it is shown that each at most countable structure of a finite signature, which is $\Sigma$-definable over $\mathbb{HF}(\mathbb R)$ with at most countable equivalence classes and without parameters, has a computable isomorphic copy. Moreover, if we lift the requirement on the cardinalities of the classes in a definition then such a model can have an arbitrary hyperarithmetical complexity, but it will be hyperarithmetical in any case. Also it is proved that any countable structure $\Sigma$-definable over $\mathbb{HF}(\mathbb C)$, possibly with parameters, has a computable isomorphic copy and that being $\Sigma$-definable over $\mathbb{HF}(\mathbb H)$ is equivalent to being $\Sigma$-definable over $\mathbb{HF}(\mathbb R)$.