\psi }{\partial
  \widetilde{y}}(\widetilde{y},\eta )&=&\frac{2}{i}\frac{1}{2}
(\frac{\partial }{\partial \Re
  \widetilde{y}}+\frac{1}{i}\frac{\partial }{\partial \Im
  \widetilde{y}})
(-\eta \cdot \Im \widetilde{y} +G(\Re \widetilde{y},\eta ))\\
&=& -(\frac{\partial }{\partial \Im \widetilde{y}}+i\frac{\partial
}{\partial \Re \widetilde{y}})(-\eta \cdot \Im \widetilde{y} +G(\Re
\widetilde{y},\eta ))\\
&=& \eta -i\nabla _y G(\Re \widetilde{y},\eta ).\end{aligned}$$ Hence $$(\widetilde{y},\frac{2}{i}\frac{\partial \psi }{\partial
  \widetilde{\eta }})
=(y,\eta )+iH_G(y,\eta ),$$ if we choose $y=\Re \widetilde{y}$.

Now consider an FBI (or generalized Bargmann-Segal) transform $$Tu(x;h)=h^{-\frac{3n}{4}}\int e^{i\phi (x,y)/h}a(x,y;h)u(y)u(y)dy,$$ where $\phi $ is holomorphic near $(x_0,y_0)\in {\bf C}^n\times {\bf
  R}^n$, $\Im \phi ''_{y,y}>0$, $\det \phi ''_{x,y}\ne 0$, $-\frac{\partial \phi }{\partial y}=\eta _0\in {\bf R}^n$, and $a$ is holomorphic in the same neighborhood with $a\sim a_0(x,y)+ha_1(x,y)+...$ in the space of such functions with $a_0\ne 0$. We can view $T$ as a Fourier integral operator with complex phase and the associated canonical transformation is $$\kappa =\kappa _T:\ (y,-\frac{\partial  \phi }{\partial y}(x,y))\mapsto
(x,\frac{\partial \phi }{\partial x}(x,y))$$ from a complex neighborhood of $(y_0,\eta _0)$ to a complex neighborhood of $(x_0,\xi _0)$, where $\xi _0=\frac{\partial \phi
}{\partial x}(x_0,y_0)$. Complex canonical transformations preserve the class of I-Lagrangian manifolds and (locally), $$\kappa ({\bf R}^{2n})=\Lambda _{\Phi _0}=\{
(x,\frac{2}{i}\frac{\partial \Phi _0}{\partial x}(x));\ x\in
\mathrm{neigh\,}(x_0,{\bf C}^n)\},$$ where $\Phi _0$ is smooth and strictly plurisubharmonic. Actually, [$$\label{ir.1}
\Phi _0(x)=\sup_{y\in {\bf R}^n}-\Im \phi (x,y),$$]{} where the supremum 