Sobolev spaces $H^s_{\mathrm {loc}}(M)$ on a real manifold $M$ are ical objects of Analysis. In this paper, we assume that $M$ is real analytic and denote by $M_{\mathrm {sa}} $ the associated subanalytic site, for which the open sets are the open relatively compact subanalytic subsets and the coverings are, roughly speaking, the finite coverings. For $s\in \mathbb R, s\leq 0$, we construct an object $\mathcal {H} ^s$ of the derived category $\mathsf {D}^{+}(\mathbb C _{M_{\mathrm {sa}}} )$ of sheaves on $M_{\mathrm {sa}}$ with the property that if $U$ is open in $M_{\mathrm {sa}} $ and has a Lipschitz boundary, then the object $\mathcal {H} ^s(U):= \mathrm {R}\Gamma (U;\mathcal {H} ^s)$ is concentrated in degree $0$ and coincides with the ical Sobolev space $H^s(U)$. This construction is based on the results of S. Guillermou and P. Schapira in this volume. Moreover, in the special case where the manifold $M$ is of dimension $2$, we will compute explicitly the complex $\mathcal {H} ^s(U)$ and prove that it is always concentrated in degree $0$, but is not necessarily a subspace of the space of distributions on $U$.