We study tautological rings for high-dimensional manifolds, that is, for each smooth manifold $M$ the ring $R\ast$($M$) of those characteristic classes of smooth fibre bundles with fibre $M$ which is generated by generalised Miller–Morita–Mumford classes. We completely describe these rings modulo nilpotent elements, when $M$ is a connected sum of copies of $Sn$ × $Sn$ for $n$ odd.