[JadeNB]

I'm not an expert in this area, but, precisely for the reason you mention, I'd expect it's easier to solve a polynomial inequality over ℚ by solving it over ℝ and intersecting down to ℚ, rather than by working directly over ℚ.

[macrolocal]

As long as you don’t bump into Gal(ℝ/ℚ) somewhere along the way.