Non-algebraic deformations of flat K\"ahler manifolds

Let $X$ be a compact K"ahler manifold with vanishing Riemann curvature. We prove that there exists a manifold $X^{\mathrm{\prime }}$, deformation equivalent to $X$, which is not an analytification of any projective variety, if and only if $H^0(X, \Omega^2) \neq 0$. Using this, we recover a recent theorem of Catanese and Demleitner, which states that a rigid smooth quotient of a complex torus is always projective. We also produce many examples of non-algebraic flat K"ahler manifolds with vanishing first Betti number.