  We introduce a new approach to the anisotropic Calder\'on problem, based on a
map called Poisson embedding that identifies the points of a Riemannian
manifold with distributions on its boundary. We give a new uniqueness result
for a large class of Calder\'on type inverse problems for quasilinear equations
in the real analytic case. The approach also leads to a new proof of the result
by Lassas and Uhlmann (2001) solving the Calder\'on problem on real analytic
Riemannian manifolds. The proof uses the Poisson embedding to determine the
harmonic functions in the manifold up to a harmonic morphism. The method also
involves various Runge approximation results for linear elliptic equations.
