  According to von Neumann, the global Hamiltonian of whole universe must be
Hermitian in order to keep the eigenvalues real and to construct a
self-consistent quantum theory. In addition to the open system approach by
introducing environmental degrees of freedom to a small system, a global
Hermitian Hamiltonian can also be generated through the dilation from a small
Hilbert space. For example, a local non-Hermitian $\cal PT$-symmetric system
can be simulated with a global Hermitian one by the Naimark dilation. When
shared by Alice and Bob, the internal nonlocality in such dilated Hermitian
systems is revealed recently, but only with a two-fold structure. In this
paper, we extend such a discussion to the generalized case when the two-fold
structure breaks. The internal nonlocality is discussed with different
correlation pictures and the corresponding correlation bounds. Our results
provide a device-independent test on the reliability of the simulation in the
global Hermiticity.
