We revisit and extend the construction of six-dimensional orientifolds built upon the T4/ℤN orbifolds with a non-vanishing Kalb-Ramond background, both in the presence of \( \mathcal{N} \) = (1, 0) supersymmetry and Brane Supersymmetry Breaking, thus amending some statements present in the literature. In the N = 2 case, we show how the gauge groups on unoriented D9 and D5 (anti-)branes do not need to be correlated, but can be independently chosen complex or real. For N > 2 we find that the Diophantine tadpole conditions severely constrain the vacua. Indeed, only the N = 4 orbifold with a rank-two Kalb-Ramond background may admit integer solutions for the Chan-Paton multiplicities, if the ℤ4 fixed points support O5− planes, both with and without supersymmetry. All other cases would involve a fractional number of branes, which is clearly unacceptable. We check the consistency of the new ℤ2 and ℤ4 vacua by verifying the unitarity constraints for string defects coupled to Ramond-Ramond two-forms entering the Green-Schwarz-Sagnotti mechanism.
