Surreality is a characteristic of being extremely disorienting and bizarre. The surreality of seeing your dog pedaling a unicycle will make you realize you're having that strange dream again.
Surreality comes from Surrealism, a European art movement that started after World War I and is known for images like Salvador Dali's painting of melting watches. The word itself means "beyond realism" in French, and surreality is distinguished by the sense that things seem real but are so illogical and unsettling that they can't possibly be. Sometimes everyday life seems like a bizarre dream, its surreality making you pinch yourself to see if you're actually asleep.