On the Kodaira problem for uniruled K\"ahler spaces

We discuss the Kodaira problem for uniruled K"ahler spaces. Building on a construction due to Voisin, we give an example of a uniruled K"ahler space $X$ such that every run of the $K_X$-MMP immediately terminates with a Mori fibre space, yet $X$ does not admit an algebraic approximation. Our example also shows that for a Mori fibration, approximability of the base does not imply approximability of the total space.