Let $x$ be a real point of a complex variety $V_\mathbb C \subseteq \mathbb C^n$ that is smooth of dimension $dim(V_\mathbb C)=m$ and defined by polynomials over the real numbers. Then we can always choose exactly $n-m$ elements $f_1,\dots, f_{n-m}$ of the ideal of $V_\mathbb C$ whose derivatives together generate the conormal space to $V_{\mathbb C}$ at $x$. The complex vanishing set of $f_1,\dots, f_{n-m}$ is smooth of dimension $m$ at $x$ by the complex intermediate value theorem, and the real vanishing set of $f_1,\dots, f_{n-m}$ is smooth of dimension $m$ in a neighborhood of $x$ by the real intermediate value theorem.

If $V_{\mathbb R}$ is not smooth of dimension $m$ at $x$, then it is a proper subset of the real vanishing set of $f_1,\dots, f_{n-m}$ in every neighborhood of $x$, so $V_\mathbb C$ is a proper subset of the complex vanishing set of $f_1,\dots, f_{n-m}$ in every neighborhood of $x$, contradicting that $V_\mathbb C$ is smooth of dimension $m$ at $x$.

Let $x$ be a real point of a complex variety $V_\mathbb C \subseteq \mathbb C^n$ that is smooth of dimension $\dim(V_\mathbb C)=m$ and defined by polynomials over the real numbers. Then we can always choose exactly $n-m$ elements $f_1,\dotsc, f_{n-m}$ of the ideal of $V_\mathbb C$ whose derivatives together generate the conormal space to $V_{\mathbb C}$ at $x$. The complex vanishing set of $f_1,\dotsc, f_{n-m}$ is smooth of dimension $m$ at $x$ by the complex intermediate value theorem, and the real vanishing set of $f_1,\dotsc, f_{n-m}$ is smooth of dimension $m$ in a neighborhood of $x$ by the real intermediate value theorem.

If $V_{\mathbb R}$ is not smooth of dimension $m$ at $x$, then it is a proper subset of the real vanishing set of $f_1,\dotsc, f_{n-m}$ in every neighborhood of $x$, so $V_\mathbb C$ is a proper subset of the complex vanishing set of $f_1,\dotsc, f_{n-m}$ in every neighborhood of $x$, contradicting that $V_\mathbb C$ is smooth of dimension $m$ at $x$.

Let $x$ be a real point of a complex variety $V_\mathbb C \subseteq \mathbb C^n$ that is smooth of dimension $n$ and defined by polynomials over the real numbers. Then we can always choose exactly $n-m$ elements $f_1,\dots, f_{n-m}$ of the ideal of $V_\mathbb C$ whose derivatives together generate the conormal space to $V_{\mathbb C}$ at $x$. The complex vanishing set of $f_1,\dots, f_{n-m}$ is smooth of dimension $m$ at $x$ by the complex intermediate value theorem, and the real vanishing set of $f_1,\dots, f_{n-m}$ is smooth of dimension $m$ in a neighborhood of $x$ by the real intermediate value theorem.

If $V_{\mathbb R}$ is not smooth of dimension $m$ at $x$, then it is a proper subset of the real vanishing set of $f_1,\dots, f_{n-m}$ in every neighborhood of $x$, so $V_\mathbb C$ is a proper subset of the complex vanishing set of $f_1,\dots, f_{n-m}$ in every neighborhood of $x$, contradicting that $V_\mathbb C$ is smooth of dimension $m$ at $x$.

Let $x$ be a real point of a complex variety $V_\mathbb C \subseteq \mathbb C^n$ that is smooth of dimension $dim(V_\mathbb C)=m$ and defined by polynomials over the real numbers. Then we can always choose exactly $n-m$ elements $f_1,\dots, f_{n-m}$ of the ideal of $V_\mathbb C$ whose derivatives together generate the conormal space to $V_{\mathbb C}$ at $x$. The complex vanishing set of $f_1,\dots, f_{n-m}$ is smooth of dimension $m$ at $x$ by the complex intermediate value theorem, and the real vanishing set of $f_1,\dots, f_{n-m}$ is smooth of dimension $m$ in a neighborhood of $x$ by the real intermediate value theorem.

If $V_{\mathbb R}$ is not smooth of dimension $m$ at $x$, then it is a proper subset of the real vanishing set of $f_1,\dots, f_{n-m}$ in every neighborhood of $x$, so $V_\mathbb C$ is a proper subset of the complex vanishing set of $f_1,\dots, f_{n-m}$ in every neighborhood of $x$, contradicting that $V_\mathbb C$ is smooth of dimension $m$ at $x$.