I just came across my own question again and think meanwhile I can give an answer:
By definition (e.g. Definition 8.3.4 in sheaves on manifolds by M. Kashiwara and P. Schapira), a sheaf $G\in D^b(M)$ is $\mathbb{R}$-constructible if
- there is a locally finite covering $M=\cup_\alpha M_\alpha$ of $M$ by subanalytic subsets such that $H^i(G)\vert_{M_\alpha}$ is locally constant for any $\alpha$ and $i$, and
there is a locally finite covering $M=\cup_\alpha M_\alpha$ of $M$ by subanalytic subsets such that $H^i(G)\vert_{M_\alpha}$ is locally constant for any $\alpha$ and $i$, and
for any $x\in M$, $G_x$ is a perfect complex.
- for any $x\in M$, $G_x$ is a perfect complex.
In particular, for $G\in D^b_{\mathbb{R}c}(M)$ we have $G_U\in D^b_{\mathbb{R}c}(M)$ for the open subanalytic $U\subset M$ in question. Now set $G=j_!F\in D^b_{\mathbb{R}c}(M)$, then we have $D_MG\in D^b_{\mathbb{R}c}(M)$ and so $$j_!(D_UF)\simeq j_!(D_Uj^{-1}G)\simeq j_!(j^{-1}D_MG)\simeq (D_MG)_U\in D^b_{\mathbb{R}c}(M).$$
