  In this paper we develop the definition of a global orthogonal spectrum and
its unitary version. It relates $G-$equivariant spectra by equivariant weak
equivalence in a coherent way. This category of global spectra has a model
structure Quillen equivalent to the global model structure on orthogonal
spectra. We also show that there is a large family of equivariant cohomology
theories, including quasi-elliptic cohomology, that can be globalized in the
new context. Starting from one global ring spectrum, we can construct
infinitely many distinct global ring spectra. Moreover, in light of the results
in this paper, we ask whether we have the conjecture that the globalness of a
cohomology theory is completely determined by the formal component of its
divisible group and when the $\acute{e}$tale component of it varies the
globalness does not change.
