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Theorem 4.2.5 of Duistermaat's "Fourier Integral Operators", 1996, states:

Let $A \in I^m(X,Y,C)$ be an elliptic Fourier Integral Operator of order $m$, associated to a bijective canonical homogeneous transformation $C$ from an open cone $\Gamma \subset T^\ast Y \setminus 0$ into $T^\ast X \setminus 0$. Then for any closed (in $T^\ast Y \setminus 0$) cone $\Gamma_0 \subset \Gamma$ such that $C(\Gamma_0)$ is closed in $T^\ast X \setminus 0$ one can find a properly supported Fourier Integral Operator $B \in I^{-m}(Y,X,C^{-1})$ such that for any $u \in \mathscr E'(Y)$, for any $v \in \mathscr E'(X)$ the following relations are valid: $$ WF(BAu - u) \cap \Gamma_0 = \varnothing,\quad WF(ABv - v) \cap C(\Gamma_0) = \varnothing. $$

The proof of the theorem involves multiplications by smooth functions with compact supports. This is why I can't directly generalize the proof of the abovementioned theorem to the case of the real-analytic category (the only real analytic function with compact support is zero). On the other hand, I think that the theorem is still valid for real-analytic Fourier Integral Operators and for analytic wavefronts but uses some more subtle proof. Please tell me, if you have encountered the mentioned theorem (or some variants) in the real-analytic case in some paper.

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You may be able to read the Sato-Kawai-Kashiwara lecture notes if your algebraic geometry background is sufficient for this non-trivial task.

On the other hand, the book by J. Sjöstrand "Singularités analytiques microlocales" in the Astérisque series, is offering an approach to analytic Fourier integral operators which is more friendly to analysts, since it is based upon the Fourier-Bros-Iagolnitzer transform which is simply a convolution by a complex Gaussian. The analytic wave-front-set gets characterized rather simply via the FBI transform and the FIO theory follows lines quite similar to the smooth case.

Another reference with a similar perspective is F. Treves' book on pseudodifferential operators and FIO edited by Plenum Press.

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