This contribution considers nonlinear space-time equalisation (STE) for multiple receive-antenna induced single-input multiple-output (SIMO) systems. By exploiting the inherent symmetry of the underlying optimal Bayesian STE solution, a novel symmetric radial basis function (RBF) based STE scheme is proposed, which is capable of approaching the optimal Bayesian equalisation performance. Adaptive implementation of this symmetric RBF (SRBF) based STE can be achieved conveniently by estimating the SIMO channels using the least mean square channel estimator and computing the optimal RBF centres from the resulting SIMO channel matrix estimate. Simulation results also demonstrate that the performance of this SRBF based STE is robust with respect to the choice of the RBF variance value. The proposed adaptive solution is then extended to the space-time decision feedback equalisation (ST-DFE) structure.