  In this article we study the linearized anisotropic Calder\'on problem on a
compact Riemannian manifold with boundary. This problem amounts to showing that
products of pairs of harmonic functions of the manifold form a complete set. We
assume that the manifold is transversally anisotropic and that the transversal
manifold is real analytic and satisfies a geometric condition related to the
geometry of pairs of intersecting geodesics. In this case, we solve the
linearized anisotropic Calder\'on problem. The geometric condition does not
involve the injectivity of the geodesic X-ray transform. Crucial ingredients in
the proof of our result are the construction of Gaussian beam quasimodes on the
transversal manifold, with exponentially small errors, as well as the FBI
transform characterization of the analytic wave front set.
