It is known that, in general, the sheaf of real analytic functions on a real analytic manifold is not coherent. However, there are some examples, where we have the coherence: for example, if $X$ is a complex analytic manifold, then the sheaf of real analytic functions $A^{\omega}_X$ is the restriction to the diagonal of the sheaf $\mathcal O_{X \times \bar X}$ of complex analytic functions on the product of $X$ and its complex conjugate $\bar X$, so $A^{\omega}_X$ is coherent.

Are there other examples of real analytic manifolds for which we can prove the coherence of the structure sheaf?

I am especially interested in the case where $X$ has the metric with special holonomy, for example, when it is a $G_2$ manifold.

It is known that, in general, the sheaf of real analytic functions on a real analytic manifold is not coherent. However, there are some examples, where we have coherence: for example, if $X$ is a complex analytic manifold, then the sheaf of real analytic functions $A^{\omega}_X$ is the restriction to the diagonal of the sheaf $\mathcal O_{X \times \bar X}$ of complex analytic functions on the product of $X$ and its complex conjugate $\bar X$, so $A^{\omega}_X$ is coherent.

Are there other examples of real analytic manifolds for which we can prove the coherence of the structure sheaf?

I am especially interested in the case where $X$ has the metric with special holonomy, for example, when it is a $G_2$ manifold.