Use the Whitney embedding theorem to realize the smooth manifold as the zero locus of non-degenerate smooth functions, then use the Weierstrass approximation theorem? to replace these by polynomials. The resulting zero locus gives the desired real-analytic version of the manifold.

One can also use Whitney’s methods to show that two diffeomorphic real analytic manifolds are real-analytic equivalent provided that they both admit embeddings in Euclidean space, equivalently provided that global, real-valued, real analytic functions distinguish points. This preliminary result also holds true for any other category of functions and manifolds, or Nash category, with an implicit and inverse function theorem and a few other features. For example, it’s true for Nash manifolds. The hard part is then showing that every real analytic manifold does indeed embed in Euclidean space. This is the Morrey-Grauert theorem. Grauert’s proof, the more sophisticated one, takes as its starting point a complex analytic manifold which is a tubular neighborhood of the real analytic manifold. It then applies some hard stuff in several complex variables to prove the existence of many global, complex analytic functions on the complex manifold.

Johannes Huisman, The exponential sequence in real algebraic geometry and Harnack’s Inequality for proper reduced real schemes, Communications in Algebra, Volume 30, Issue 10, 2002 (pdf)

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