On the defect in the generalized Grunwald--Wang problem

The classical Grunwald--Wang theorem asserts that, unless we are in the so-called special case, local cyclic Galois extensions at finitely many completions of a number field can be approximated by a global cyclic extension. In the special case the obstruction is measured by a group of order 2. It has been known for a long time that the Grunwald--Wang theorem extends to a very general context of valued fields. Therefore it is natural to ask whether in the special case the obstruction is always measured by a finite group and if so, is the order of this group bounded independently of the number of places under consideration. We show that the answer to both questions is negative in general, already for rational function fields and discrete valuations coming from points of the affine line. This has some interesting links to the arithmetic of function fields over Q or Q_p.