The ontological meaning of the wave function has been debated for many years. A widely-discussed view is wave function realism, according to which the wave function represents a real physical field in a fundamental high-dimensional space. In this paper, I argue that wave function realism is inconsistent with the reality of the global phase. In other words, the global phase cannot be real in wave function realism, and if the global phase is real, then wave function realism will be not true. Moreover, if the global phase is real, then the wave function of an $N$-body system cannot represent a single physical entity, either in a high-dimensional space or in three-dimensional space, but represent $N$ physical entities in three-dimensional space.