  Given a conformally transversally anisotropic manifold $(M,g)$, we consider
the semilinear elliptic equation
  $$(-\Delta_{g}+V)u+qu^2=0\quad \text{on $M$}.$$ We show that an a priori
unknown smooth function $q$ can be uniquely determined from the knowledge of
the Dirichlet-to-Neumann map associated to the semilinear elliptic equation.
This extends the previously known results of the works [FO20, LLLS21a]. Our
proof is based on analyzing higher order linearizations of the semilinear
equation with non-vanishing boundary traces and also the study of interactions
of two or more products of the so-called Gaussian quasimode solutions to the
linearized equation.
