(KS = Kashiwara–Schapira, Sheaves on Manifolds.)
Let $f:X\to\mathbb C$ be holomorphic and $F$ a $\mathbb C$-constructible complex (all functors derived). Fix $y\in Y:=f^{-1}(0)$, choose a Milnor ball $B$ at $y$ and a disk $D$ about $0$ with $f(B)\subset D$ and $0$ the only critical value in $D$; set $D^\ast=D\setminus\{0\}$. Write
$$ \phi_\theta=\Re(e^{-i\theta}f),\quad \psi_\theta=\Im(e^{-i\theta}f),\quad u_\theta:\{\phi_\theta>0\}\hookrightarrow X. $$
By KS Prop. 8.3.12 there is a finite $\Sigma$ (uniform on $K:=B\cap f^{-1}(\overline{D^\ast})$) such that for $\theta\notin\Sigma$, $d\phi_\theta$ is non-characteristic on $B\cap f^{-1}(D^\ast)$; since $\mathrm{SS}(F)$ is $J$-invariant, also $d\psi_{\theta\pm\delta}$ are non-characteristic for small $\delta$. Consider
$$ S_{\theta,\delta}=\{\psi_{\theta+\delta}>0\}\cap\{\psi_{\theta-\delta}<0\}\subset D^\ast. $$
Working in an open Milnor ball (or by excision), applying KS Cor. 5.4.19 in two monotone steps—first to the wedge between the two $\psi$-level sets, then to $\{\phi_\theta>0\}$—gives canonical isomorphisms
$$ R\Gamma\!\bigl(B\cap f^{-1}(D^\ast);F\bigr)\ \simeq\ R\Gamma\!\bigl(B\cap f^{-1}(S_{\theta,\delta});F\bigr)\ \simeq\ R\Gamma\!\bigl(B\cap\{\phi_\theta>0\};F\bigr), $$
hence near $y$,
$$ \psi_f^{\mathrm{na}}(F)=i^\ast j_\ast j^{\ast}F\ \simeq\ i^\ast u_{\theta\ast}u_\theta^{\ast}F \qquad(\text{generic }\theta). $$
Question. Is this application of KS Cor. 5.4.19 to the angular deformation (between two non-characteristic $\psi$-level sets, then to $\{\phi_\theta>0\}$) valid as stated? If not, what additional local hypothesis is needed?