Suppose $X$ is the set of $\mathbb R$-points of a scheme $\mathfrak X$. Let $x\in \mathfrak X$ be a simple point with a smooth neighbourhood $\mathfrak U$. I presume the (scheme-theoretic) tangent space at $x\in\mathfrak U$ coincides with the (differential-geometric) tangent space at $x\in U$, where $U$ denotes the set of $\mathbb R$-points of $\mathfrak U$ with a manifold structure.
Do you know of a reference for this? (If the statement is true, that is, and I haven't failed to assume algebraic closure or something like that.)