The term "Red Team" first originated from US Army training exercises in the Cold War era, in which an opposing force (OPFOR) would challenge a unit in training scenarios using weapons and tactics that they were likely to face. Since the US was most likely to face Russian units, the OPFOR was known as the "Red Team", while the allied (US/NATO) team was known as the "Blue Team".
The last name of the person stuck in the reverie (Hallo) is likely an homage to the immersive Halo: Combat Evolved (2001) series of first-person shooter video games.