May I suggest that before we even attempt to think about the (likely very delicate) question of constructive quantifier elimination for the reals and/or for real-closed field (and to what extent $\mathbb{R}$ is real-closed or what this even means constructively), the first step should be to fully understand the analogous but simpler questions about $\mathbb{C}$ (and the language consisting of $+,\times,\#$ where $\#$ means “apart”) and Chevalley's theorem about projection of algebraically constructible sets.
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