  Let $k$ be a number field and let $T$ be a $k$-torus. Consider a fibration in
torsors under $T$, i.e. a morphism $f: X \to \mathbb{P}^1_k$ from a smooth,
projective $k$-variety $X$ to $\mathbb{P}^1_k$ such that the generic fibre
$X_\eta \to \eta$ is a smooth compactification of a principal homogeneous space
under $T \times_k \eta$.
  We study the Brauer-Manin obstruction to the Hasse principle and weak
approximation for $X$, under Schinzel's hypothesis, thereby generalizing recent
work of Wei. Our results are unconditional if $k = \mathbb{Q}$ and the
non-split fibres of $f$ are defined over $\mathbb{Q}$.
