  We show that a continuous potential $q$ can be constructively determined from
the knowledge of the Dirichlet-to-Neumann map for the Schr\"odinger operator
$-\Delta_g+q$ on a conformally transversally anisotropic manifold of dimension
$\geq 3$, provided that the geodesic ray transform on the transversal manifold
is constructively invertible. This is a constructive counterpart of the
uniqueness result of Dos Santos Ferreira-Kurylev-Lassas-Salo. A crucial role in
our reconstruction procedure is played by a constructive determination of the
boundary traces of suitable complex geometric optics solutions based on
Gaussian beams quasimodes concentrated along non-tangential geodesics on the
transversal manifold, which enjoy uniqueness properties. This is achieved by
applying the simplified version of the approach of Nachman-Street to our
setting. We also identify the main space introduced by Nachman-Street with a
standard Sobolev space on the boundary of the manifold. Another ingredient in
the proof of our result is a reconstruction formula for the boundary trace of a
continuous potential from the knowledge of the Dirichlet-to-Neumann map.
