Both marginal and dependence features must be described when modelling the extremes of a stationary time series. There are standard approaches to marginal modelling, but long-and short-range dependence of extremes may both appear. In applications, an assumption of long-range independence often seems reasonable, but short-range dependence, i.e., the clustering of extremes, needs attention. The extremal index 0 < theta <= 1 is a natural limitingmeasure of clustering, but for wide classes of dependent processes, including all stationary Gaussian processes, it cannot distinguish dependent processes from independent processes with theta = 1. Eastoe and Tawn (Biometrika 99, 43-55 2012) exploit methods from multivariate extremes to treat the subasymptotic extremal dependence structure of stationary time series, covering both 0 < theta < 1 and theta = 1, through the introduction of a threshold-based extremal index. Inference for their dependence models uses an inefficient stepwise procedure that has various weaknesses and has no reliable assessment of uncertainty. We overcome these issues using a Bayesian semiparametric approach. Simulations and the analysis of a UK daily river flow time series show that the new approach provides improved efficiency for estimating properties of functionals of clusters.