Let $G$ be a Roelcke-precompact non-archimedean Polish group, $\mathcal{B}(G)$ the algebra of matrix coefficients of $G$ arising from its continuous unitary representations. The Gel’fand spectrum $H(G)$ of the norm closure of $\mathcal{B}(G)$ is known as the Hilbert compactification of $G$. Let $\mathcal{A}_G$ be the dense subalgebra of $\mathcal{B}(G)$ generated by indicator maps of open cosets in $G$. We prove that multiplicative linear functionals on $\mathcal{A}_G$ are automatically continuous, generalizing a result of Krein for finite dimensional representations of topological groups. We deduce two abstract realizations of $H(G)$. One is the space $P(\mathcal{M}_G)$ of partial isomorphisms with algebraically closed domain of $\mathcal{M}_G$, the countable set of open cosets of $G$ seen as a homogeneous first order logical structure. The other is $T(G)$ the Tannaka monoid of $G$. We also obtain that the natural functor that sends $G$ to the category of its representations is full and faithful.