In electromagnetism, while the Maxwell equations are time symmetric, there is a choice to restrict solutions specifically to retarded potentials, imposing a time direction on the equations. And in QFT, as far as I'm aware, there is a restriction of what constitute an acceptable field to the small groups with a forward in time pointing momentum (in Weinberg at least).
On the other hand, the propagator usually used in QFT is the Feynman propagator, which is time symmetric. And the axiomatisations of QFT do not seem to include any specific axiom regarding this. Wightman just requires operators to commute outside the lightcone, but doesn't seem to differentiate the two lightcones. Local QFT requires spacetime to be time oriented, but I am not quite sure in what axiom this is actually used, as the category of regions is over the double lightcones.
Is QFT defined to be time symmetric or is there a direction of time assumed? Is that some implicit assumption, or does it derive from the axioms of QFT? Or else what is to prevent to have two different fields from having momentums in different directions?