Measurements, QFT and Wightman's axiom 3

I think I might have misconceptions about the conceptual core of QFT. Let me explain where I am puzzled.

In QM, the measurement process is accounted by the postulate of collapse of the wave function: when a measurement is made on a state, the unit vector defining the state is projected on one of the eigenvectors of the basis defining the observable; now, (let us assume that the Hilbert space is finite-dimensional) two self-adjoint commute if and only if they have a common basis of eigenvectors. So, commuting self-adjoint operators model observables that can be simultaneously measured.

Let me now assume that the quantum fields, in quantum field theories, are to be thought as observables on spacetime, and that the measurement of an observable yields to a wave function collapse; I understand why one would ask that two fields with space-like separated supports should commute. But Wightman’s axiom 3 also allows two such fields to be anti-commuting. What would happen if two space-like separated observers repeatedly made the same measurements (each one with respect to a field, and such that the two fields would anti-commute)? Would they randomly see their results switch sign, once in a while, thus telling them that the other has made his/her measurement? Wouldn’t this imply super-luminal communication?

So, where am I wrong?