Do there exist two Noetherian integral domains over $\mathbb{C}$ whose $\mathbb{C}$-tensor product is Noetherian such that neither of them is a localization of a finitely generated $\mathbb{C}$-algebra?
The motivating example is $\mathbb{C}[[x]]\otimes \mathbb{C}((x))$ which is a non-Noetherian subalgebra of $\mathbb{C}((x))[[x]]$. Here, neither of the factors is a localization of a finitely generated algebra and the product is not Noetherian.