  We describe three previously unnoticed components of the moduli space of
minimally supersymmetric string theories in $d\geq 7$, describing in some
detail their spectrum and duality properties. We find a new component in nine
and eight dimensions, and two additional ones in seven dimensions. These
theories were originally discovered in a bottom-up classification of possible
F/M-theory singularity freezing patterns in the K3 lattice, described in a
companion paper. The 9d/8d component can be understood as F/M-theory on a
twisted fibration of the Klein bottle over a circle, while the new
seven-dimensional components are described as IIB on Bieberbach manifolds with
a duality bundle and RR-NSNS backgrounds turned on. All the new components can
be obtained from previously known theories by turning on certain discrete theta
angles; however, the spectrum of massive objects is very different, and most
strikingly, they feature an incomplete lattice of BPS strings, showing that
string BPS completeness is not true in general even with sixteen supercharges.
In all cases we find non-BPS representatives for each value of the charge, so
the Completeness Principle is satisfied. We also analyze analogous theta angles
in nonsupersymmetric string theories, and provide a detailed explanation of why
the Type I discrete $\theta$ angle proposed in 1304.1551 is unphysical, using
this to clarify certain non-perturbative phenomena in $O8$ planes.
