  In this note we show that on any compact subdomain of a K\"ahler manifold
that admits sufficiently many global holomorphic functions, the products of
harmonic functions form a complete set. This gives a positive answer to the
linearized anisotropic Calder\'on problem on a class of complex manifolds that
includes compact subdomains of Stein manifolds and sufficiently small
subdomains of K\"ahler manifolds. Some of these manifolds do not admit limiting
Carleman weights, and thus cannot by treated by standard methods for the
Calder\'on problem in higher dimensions. The argument is based on constructing
Morse holomorphic functions with approximately prescribed critical points. This
extends results of Guillarmou and Tzou (Duke Math. J. 2011) from the case of
Riemann surfaces to higher dimensional complex manifolds.
