Split brain experiments involve studying rare participants who have severed connections between hemispheres via the corpus callosum. Roger Sperry (Another Nobel Prize winner, Beran wins the day!) was the first to study these patients (later, Mike Gazzaniga), and reported interesting phenomena that appeared to show each hemisphere having separate perceptual experiences that then led to different decisions or responses. Often, these people would provide verbal responses that suggested a clear conflicts between hemispheres in terms of information processing. For example, a patient with a split brain is shown a picture of a chicken foot and a snowy field in separate visual fields and asked to choose from a list of words the best association with the pictures. The patient would choose a chicken to associate with the chicken foot and a shovel to associate with the snow; however, when asked to reason why the patient chose the shovel, the response would relate to the chicken (e.g. "the shovel is for cleaning out the chicken coop").[I took this last example from Wikipedia]
For a long time, there was the argument that these kinds of phenomena suggested "two consciousnesses" in the brain, and one that (because it lacked access to language outputs) was largely muted but still present. However, more recently, this particular idea has been challenged by new evidence that suggests split hemispheres still produce a unitary state of consciousness.