Real-booting is the natural ability of Gigalomaniacs and the artificial ability of Nozomi's "Noah II" to falsefuly trick a person into thinking delusions are real.("turn delusions into reality" - is not a correct statement, as something hallucinated is not reality, these are opposing ideals, its a contradicted statement).
The character Aoi Sena explains that every operation in the world can be described using the numbers 0, 1, and -1. By using a Di-sword to access the Dirac Sea and create pair production leading to the occurance of particles and antiparticles, Gigalomaniacs are able to give substance to delusions.
By pushing generated particles or energy waves into the dead/blind spot in the minds of people and letting them undergo shared recognition by several people, the particles or delusions are manifested into "true reality". In other words delusions are "proved" real, because the are perceived as real or truth or part of reality by more than one observer. If they are not perceived through shared recognition by surrounding people then they have no substance and are simply delusional "errors"
Errors that are perceived by multiple Gigalomaniacs are not, however, given substance for anyone else but the Gigalomaniacs themselves, only by "real-booting" will they truly gain substance and be able to effect everyone.
"Real-booting" can be used to create as well as disguise beings and objects, even feelings can be created. In contrast by pushing antiparticles into a person's dead/blind spot the manifested delusion is negated, thereby "freeing" the perceiver from the delusion which was made real.
Delusion Synchronization is caused when one or more gigalomaniacs synchronizes their own delusion with that of another. By doing so they can distort the rules/delusions that others have created. This allows a gigalomaniac in combat to turn a delusion directed against them back against its owner.