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We describe a method for aligning multiple unlabeled configurations simultane- ously. Specifically, we extend the two-configuration matching approach of Green and Mardia (2006) to the multiple configuration setting. Our approach is based on the in- troduction of a set of hidden locations underlying the observed configuration points. A Poisson process prior is assigned to these locations, resulting in a simplified formu- lation of the model. We make use of a structure containing the relevant information on the matches, of which there are different types to take into account. Bayesian inference can be made simultaneously on the matching and the relative transformations between the configurations. We focus on the particular case of rigid-body transformations and Gaussian observation errors. We apply our method to a problem in chemoinformatics: the alignment of steroid molecules. Supplementary materials are available online.