Question: Is the ring of real-analytic functions on $\mathbb{C}\mathbb{P}^n$ (real valued) a Noetherian ring?

References or counterexamples are welcome.

I know that the ring of germs of holomorphic functions on $\mathbb{C}^n$ is Noetherian, but the ring of holomorphic functions on $\mathbb{C}^n$ is not! I also know how to verify that the ring of polynomials is Noetherian.