Let $G$ be a connected reductive algebraic group over $\mathbf{Q}$. I've seen two slightly different definitions in the literature of the Shimura variety of level $U$, for $U \subseteq G(\mathbf{A}_{\mathrm{f}})$ an open compact subgroup.
Milne's "Introduction to Shimura varieties", following Deligne's "Travaux de Shimura", defines $$Sh_G(U) = G(\mathbf{Q})\, \backslash\, G(\mathbf{A})\, /\, U K_\infty$$ where $K_\infty \subseteq G(\mathbf{R})$ is the centraliser of the cocharacter $h: \mathbf{S} \to G_{\mathbf{R}}$ defining the Shimura datum. Deligne shows in Prop 2.6 of "Travaux de Shimura" that $$K_\infty = (\text{max compact subgroup of $G'(\mathbf{R})^\circ$}) \cdot Z_G(\mathbf{R}),$$ where $G' = $derived subgroup, and "$\circ$" denotes "identity component".
However, I've also seen a definition using a slightly different recipe: $$\widetilde{Sh}_G(U) = G(\mathbf{Q})\, \backslash\, G(\mathbf{A})\, /\, U \widetilde{K}_\infty$$ where $$\widetilde{K}_\infty = \text{(max compact subgroup of $G(\mathbf{R})^\circ)$} \times Z_G(\mathbf{R})^\circ.$$ E.g. this definition is the one used in Emerton's "On the interpolation of systems of Hecke eigenvalues" (Inventiones 2006).
These definitions really seem to be different in general. If $G$ is semisimple and adjoint (so $Z_G = \{1\}$), they are the same. If $G$ is a torus, then they differ by the component group of $G(\mathbf{R})$; in fact Milne's lecture notes mention something equivalent to $\widetilde{Sh}_G(U)$ in the setting of tori. [With Milne's definitions, a "0-dimensional Shimura variety" is not the same as a Shimura variety which happens to be 0-dimensional!]
For general $G$ it's not obvious to me if there's even a map between the two. EDIT: If I'm not mistaken, we can conjugate $K_\infty$ and $\tilde K_\infty$ so they have the same Lie algebra; $\tilde K_\infty$ is automatically connected, although $K_\infty$ may not be, so $\tilde K_\infty$ is always contained in $K_\infty$. Thus $\widetilde{Sh}_G(U)$ is a finite covering of $Sh_G(U)$.
Is it true that the compatible system of varieties $\widetilde{Sh}_G(U)$ has a model over the reflex field?
(I'm having trouble coming up with "interesting" examples where $K_\infty \ne \tilde{K}_\infty$ -- the only examples I can think of are tori, where the existence of models is trivial anyway. This question is about trying to find interesting examples.)
