Ok I think I figured out where all the confusion is coming from in regard to this.
Free adiabatic expansion of an ideal gas means the ideal gas is allowed to expand into a preset volume container, such as the one shown here:
When I initially read this thread, I thought the container was expanding (as in a piston), but that is not the case. The volume of the CONTAINER remains constant, while the volume of the GAS (real or ideal) changes. Sorry if I'm the only one just now realizing this, but it makes things very clear for me so I imagine some of you are encountering the same confusion I was.
The reason it is irreversible is because of entropy. You are not going to get 10 particles to squeeze into a 10mL space if they all have access to a 100mL space unless you put in some energy. Adiabatic = no energy input, so your reaction is not going to reverse to the original state barring being in some universe where entropy is disfavored.
No work is done by the gas because it is simply increasing entropy spontaneously and under no force or energy absorption.
So to answer the OP. Free adiabatic expansion of ideal gas does involve constant temperature. Free adiabatic expansion of real gases will involve a temperature decrease.