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Monte Carlo simulations are important pedagogical and research tools in statistical physics. Cluster methods, like Wolff, continue to be important research tools. This paper takes a method little-known to even those as immersed in the subject as Jon Machta for extending cluster algorithms to operate in a field, generalizes it to a broad class of systems and fields, and makes a quantitative test of its efficiency. This fills an much-needed gap in a canonical method. Fast simulations of the Ising model in a field have been essential in recent work on information geometry, universal scaling of correlation functions, and more.