The topological $\theta$-term in the Schwinger model (1+1-dimensional QED) can be interpreted as a background electric field, as explained in Chapter 7.1.2 of Tong's lecture notes. The same holds true for 1+1-dimensional QCD, as pointed out in this paper by Witten.
Now my question is: (how) does this change when we go to 3+1 dimensions?
I've read in several papers (e.g. by Lüscher or Dvali) that the $\theta$-term in 4-dimensional QCD gives rise to an (electric) non-Abelian background field. If this statement is true, why don't we get a non-Abelian magnetic background field as well? What happens if we go to compact 4-dimensional QED, which also exhibits a physical $\theta$-term, as pointed out in this paper by Polyakov?