Do half integer spin fields commute or anti-commute with spin integer fields?

What are the fundamental commutation/anti-commutation relations between half integer and integer spin fields? For instance, in QED do we have $$[\psi(x),A^{\mu}(y)]=0$$ or $$\lbrace\psi(x),A^{\mu}(y)\rbrace=0$$ ?

• Related (duplicate?): physics.stackexchange.com/q/235589 In particular, though the question isn't exactly the same, the first answer seems to address yours. – Chris Jan 30 '18 at 6:50

They commute; see e.g. this discussion of the $\mathbb{Z}_2$ grading involved.