An example of a non-geometric $C^\infty(M)$-module

Let $M$ be the unit circle in $\mathbb C$, and consider the algebra homomorphism $C^\infty(M)\to M_2(\mathbb R)$ given by $$ f\mapsto \begin{bmatrix} f(1) & \frac{df}{d\theta}(1) \\ 0 & f(1)\end{bmatrix}$$ (where $\theta$ is the angular coordinate on $M$). This homomorphism makes $\mathbb R^2$ into a $C^\infty(M)$-module with $\bigcap_p \mu_p \mathbb R^2 = \left[\begin{smallmatrix} \mathbb R \\ 0\end{smallmatrix}\right]$.

(In general, fix $x\in M$ and let $\mu^2_x\subset C^\infty(M)$ be the ideal of functions $f$ such that $f$ and all of its first-order derivatives vanish at $x$. Then take $Q=C^\infty(M)/\mu^2_x$. I believe that then $\bigcap_p \mu_p Q = \mu_x/\mu^2_x$.)

What about $\Gamma(M,\mathcal{C}_{M}^{\infty}/\mathcal{I}_{x}^{2})$ where $\mathcal{I}_x$ is the ideal sheaf of a point $x\in M$?