Most experiments force the choice of a singleton in each choice set, whereas the theory of choices deals with choice correspondence, i.e., assume that decision makers can choose sets of alternatives. I introduce a methodology for identifying choice correspondences experimentally. I allow decision makers to choose several alternatives, provide a small incentive for each alternative chosen, and then randomly select one for payment. I derive the conditions under which the methodology at least partially identifies the choice correspondence, by providing upper and lower bounds. I illustrate the methodology with an experiment, in which subjects chose between four paid tasks. I can retrieve the full choice correspondence for 26% of subjects and bound it for 72% of the subjects. Subjects chose sets of size 2 or larger 60% of the time, whereas only 3% of them always chose singletons. I then show that 46% of all observed choices can be rationalized by complete, reflexive and transitive preferences in my experiment, i.e., verify WARP. Going beyond WARP, I show that complete, reflexive and transitive preferences with menu-dependent choices rationalize 93% of observed choices. Incomplete preferences or just-noticeable difference preferences do not rationalize more choice correspondences than the classical model. Having elicited choice correspondences allows me to conclude that indifference is widespread in the experiment. These results show how fruitful the methodology developed can be and pave the way for exploring various behavioral models with a unified methodology.