We describe and develop a variation on a layered multishift coupler due to Wilson that uses a slice sampling procedure to allow one to obtain potentially common draws from two different distributions. Our main application is coupling sample paths of Markov chains for use in perfect sampling algorithms. The coupler is based on slicing density functions and we describe a “folding” mechanism as an attractive alternative to the accept/reject step commonly used in slice sampling algorithms. Applications of the coupler are given to storage models and to auto-gamma distribution sampling.