"In one well-known experiment, a split-brain patient’s left hemisphere was shown a picture of a chicken claw and his right hemisphere was shown a picture of a snow scene. The patient was asked to point to a card that was associated with the picture he just saw. With his left hand (controlled by his right hemisphere) he selected a shovel, which matched the snow scene. With his right hand (controlled by his left hemisphere) he selected a chicken, which matched the chicken claw. Next, the experimenter asked the patient why he selected each item. One would expect the speaking left hemisphere to explain why it chose the chicken but not why it chose the shovel, since the left hemisphere did not have access to information about the snow scene. Instead, the patient’s speaking left hemisphere replied, “Oh, that’s simple. The chicken claw goes with the chicken and you need a shovel to clean out the chicken shed”" [1]. Also [2] has an interesting hypothesis on split-brains: not two agents, but two streams of perception.
[1] 2014, "Divergent hemispheric reasoning strategies: reducing uncertainty versus resolving inconsistency", https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4204522
[2] 2017, "The Split-Brain phenomenon revisited: A single conscious agent with split perception", https://pure.uva.nl/ws/files/25987577/Split_Brain.pdf