An integral real scheme splits over the complex numbers

Given: The objects, hypotheses and conventions in the statement above.

For a field extension $K\mathrm{/}k$ and a $k$-scheme $X$, the inverse image of every affine open $U=\mathrm{Spec}A$ in $X_K$ is $\mathrm{Spec}(A{\otimes }_{k}K)$. These affine charts cover $X_K$ and are compatible on overlaps and with coefficient localizations. (Affine charts after extension of the ground field)

Let $A\to C$ be a unital ring map. For any set of variables $(t_i)$ and any ideal $I\subseteq A[t_i]$, $(A[t_i]\mathrm{/}I){\otimes }_{A}C\cong C[t_i]\mathrm{/}IC[t_i]\mathrm{.}$ Here the extended ideal is generated by the coefficient images of all elements of $I$. For a multiplicative subset $M\subseteq A$, $(M^{-1}A){\otimes }_{A}C\cong {\bar{M}}^{-1}C\mathrm{.}$ These are ring isomorphisms; no flatness, finite-generation or nonzero-ring hypothesis is required. (Presentations and localization under base extension)

Let $R$ be a commutative ring and let $I_1,\dots ,I_r$ be pairwise comaximal ideals, where $r\ge 1$. Then the canonical map $R⟶{\prod }_{i=1}^{r}R\mathrm{/}{I}_i,x⟼(x+I_1,\dots ,x+I_r)$ is surjective, its kernel is ${\bigcap }_{i=1}^r{I}_i$, and ${\bigcap }_{i=1}^r{I}_i={\prod }_{i=1}^r{I}_i\mathrm{.}$ Equivalently, $R\mathrm{/}{\prod }_{i=1}^r{I}_i\cong {\prod }_{i=1}^{r}R\mathrm{/}{I}_i\mathrm{.}$ (Chinese remainder theorem for pairwise comaximal ideals)