  We resolve a puzzle in the theory of strings propagating on locally flat
spacetimes with nontrivial Wilson lines for stringy Z_N gauge symmetries. We
find that strings probing such backgrounds are described by consistent
worldsheet CFTs. The level mismatch in the twisted sectors is compensated by
adjusting the quantization of momentum of strings winding around the Wilson
line direction in units of 1/(N^2 R) rather than 1/(N R), as might have been
classically expected. We demonstrate in various examples how this improvement
of the naive orbifold prescription leads to satisfaction of general physical
principles such as level matching and closure of the OPE. Applying our
techniques to construct a Wilson line for T-duality of a torus in the type II
string (``T-fold''), we find a new 7D solution with N=1 SUSY where the moduli
of the fiber torus are fixed. When the size of the base becomes small this
simple monodrofold exhibits enhanced gauge symmetry and a self-T-duality on the
S^1 base.
