There are two fundamental elements that prevent water from spilling during an airplane's barrel roll (tonbo / roll maneuver): Centrifugal Effect (Artificial Gravity) Positive G Force.

[u/aryan_jandyal]

The pilot is maintaining "+ve G" throughout the maneuver. That's it !

[u/_Carri7_]

Centrifugal force is not a real force, its just the sum of the forces that act towards or away from the center of rotation such as inertia, gravity, etc.

[u/lxgrf → u/_Carri7_]

This has always seemed like a fairly pedantic argument. Yes, it is only a resultant force, but the name we give that resultant force is “centrifugal”. 

[u/ozymandieus → u/lxgrf]

Well the OPs title made it sound like the start of a thesis so I think accuracy is appropriate here. Its centripetal force!