  We present a nested algebraic Bethe ansatz for a one dimensional open spin
chain whose boundary quantum spaces are irreducible $\mathfrak{so}_{2n}$- or
$\mathfrak{sp}_{2n}$-representations and the monodromy matrix satisfies the
defining relations of the Olshanskii twisted Yangian
$Y^\pm(\mathfrak{gl}_{2n})$. We use a generalization of the Bethe ansatz
introduced by De Vega and Karowski which allows us to relate the spectral
problem of a $\mathfrak{so}_{2n}$- or $\mathfrak{sp}_{2n}$-symmetric open spin
chain to that of a $\mathfrak{gl}_{n}$-symmetric periodic spin chain. We
explicitly derive the structure of the Bethe vectors and the nested Bethe
equations.
