The theorem you want is due to Serge Lang, from the following paper:

The theory of real places.
Ann. of Math. (2) 57, (1953). 378–391.

The statement is almost, but not quite, what you suggest. To see the problem, a good example to consider is the affine plane curve over $\mathbb{R}$ defined by $\mathbb{R}[x,y]/(y^2+x^2+x^4)$. This defines a geometrically integral curve over $\mathbb{R}$ with non-formally real fraction field but possessing an $\mathbb{R}$-point, namely $(0,0)$. The key is that $(0,0)$ is the only $\mathbb{R}$-point on this curve and (thus!) it is a singular point.

So the correct result is that the function field of an integral affine variety $V_{/\mathbb{R}}$ is formally real iff $V$ admits a nonsingular $\mathbb{R}$-point. (Note that projective real algebraic varieties are also affine(!!).) Probably you could extend this to finite-type integral schemes without any trouble.

I also looked in *Bochnak, Coste and Roy*, following Thierry Zell's suggestion, but only found "half" of this result, namely the Artin-Lang Homomorphism Theorem. It seems likely though that I just didn't look hard enough: perhaps someone can enlighten me.