Equilibrium Constant in terms of Molar Concentration of Gases versus Equilibrium Constant
I believe that the real $K_c$ would be higher. This is because in ideal gases london dispersion forces, hydrogen bonding, etc. is ignored. These would increase the number of collisions between particles since they would be attracted to each other more.
$$\ce k=zp*e^{-E_a/RT}$$
where k= rate constant, z= frequency of collisions, p= fractions of collisions with proper orientation to cause a reaction, $E_a$= activation energy, T= temperature. So Since z would increase, the K would increase. Hence the $K_c$ would increase.