Is it possible to extend a given function to be real analytic if its analytic wave front set consists of finitely many covectors at each point?

Question

Specifically, suppose you have a function $f:\mathbb{R}^{2} \to \mathbb{R}$ and you assume that its analytic wave front set $\mathrm{WF}_{A}(f)$ contains at most finitely many covectors $\{(x, \xi_{j})\}_{j=1}^{N_{x}} \subset \mathbb{R}^{2} \times \mathbb{S}^{1}$ at each point $x$. For simplicity, suppose that $N_{x}$ is bounded uniformly in $x$. If we assume a priori that $f$ is bounded (or even continuous), does this imply $f$ is also real-analytic? In other words, can we 'remove' the finite set $\{x, \xi_{j}\}_{j=1}^{N_{x}}$ from $\mathrm{WF}_{A}(f)$ and conclude that it is actually empty? I imagine this is similar to the notion that one can extend a bounded function analytic on the punctured plane to be analytic everywhere. However, in this case the question is dealing within phase space.