  Let $X$ be a normal compact K\"ahler space with klt singularities and torsion
canonical bundle. We show that $X$ admits arbitrarily small deformations that
are projective varieties if its locally trivial deformation space is smooth. We
then prove that this unobstructedness assumption holds in at least three cases:
if $X$ has toroidal singularities, if $X$ has finite quotient singularities,
and if the second cohomology group of its tangent sheaf vanishes.
