This has already been asked here more than once, but the existing answers do not tackle my misunderstanding.
A topological $\theta$-term is understood to be physical, in the usual particle model constructions, if it cannot be rotated away by the chiral anomaly simultaneously with a possible $\mathit{CP}$-violating mass phase. That is the case of QCD, in which an $\alpha$ chiral rotation induces a change of $2\alpha$ in $\theta$ and $-2\alpha$ in the mass phase.
Regarding the also non-abelian weak isospin $SU(2)_L$ sector, however, the situation is different. Since, distinctly from QCD, this group is chiral, i.e., only the left-handed fields measure is of consequence, it is said that the $\theta$-term can be rotated away. I do not understand this argument.
My impression is that the partition function can only acquire the $\text{tr}F\wedge F$ increment through axial rotations, i.e., transformations which rotate right and left-handed fields with opposite angle. There is no freedom, in this case, to absorb an arbitrary mass phase in the right-handed fields. What am I missing?
I have also read through standard references which imply that the anomalous symmetry which is used to remove the $\theta$-term is the baryon/lepton number (instead of the axial one), which just further confuses me.
Can someone make these statements precise?