  We build a general theory of microlocal (homogeneous) Fourier Integral
Operators in real-analytic regularity, following the general construction in
the smooth case by H\"ormander and Duistermaat. In particular, we prove that
the Boutet-Sj\"ostrand parametrix for the Szeg\H{o} projector at the boundary
of a strongly pseudo-convex real-analytic domain can be realised by an analytic
Fourier Integral Operator. We then study some applications, such as FBI-type
transforms on compact, real-analytic Riemannian manifolds and propagators of
one-homogeneous (pseudo)differential operators.
