We study solutions of the Bethe ansatz equations associated to the orthosymplectic Lie superalgebras $\mathfrak{ osp}_{ 2m+1|2n}$ and $\mathfrak{ osp}_{ 2m|2n}$. Given a solution, we define a reproduction procedure and use it to construct a family of new solutions which we call a population. To each population we associate a symmetric rational pseudo-differential operator $\mathcal R$. Under some technical assumptions, we show that the superkernel $W$ of $\mathcal R$ is a self-dual superspace of rational functions, and the population is in a canonical bijection with the variety of isotropic full superflags in $W$ and with the set of symmetric complete factorizations of $\mathcal R$. In particular, our results apply to the case of even Lie algebras of type D${ }_m$ corresponding to $\mathfrak{ osp}_{ 2m|0}=\mathfrak{ so}_{ 2m}$.
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