Non-invariance of weak approximation with Brauer--Manin obstruction

In this paper, we study weak approximation with Brauer–Manin obstruction with respect to extensions of number fields. For any nontrivial extension $L\mathrm{/}K,$ assuming a conjecture of M. Stoll, we prove that there exists a $K$-threefold satisfying weak approximation with Brauer–Manin obstruction off all archimedean places, while its base change to $L$ fails. Then we illustrate this construction with an explicit unconditional example.