Let's call $ZF$ the Zermelo-Fraenkel set theory, $AC$ the full axiom of choice, $DC$ the axiom of dependent choice and $AC_\omega$ the axiom of countable choice.
We know that $ZF$+"existence of amorphous sets" is consistent. We also know that $ZF$+$DC$+"existence of amorphous sets" is not consistent (and so $ZF$+$AC$+"existence of amorphous sets" is not consistent).
My question is the following: is $ZF$+$AC_\omega$+"existence of amorphous sets" consistent?