  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
$\mathfrak{osp}_{2m|0}=\mathfrak{so}_{2m}$.
