  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.
