  Let $K$ be a number field of degree $d\geq 3$ and fix $s$ multiplicatively
independent algebraic integers $\gamma_1, \dots, \gamma_s \in K^*$ that fulfil
some technical requirements, which can be vastly simplified to
$\mathbb{Q}$-linearly independence, given Schanuel's conjecture. We then
consider the twisted Thue equation \[
  \left|N_{K/\mathbb{Q}}\left(X-\gamma_1^{t_1}\cdots\gamma_s^{t_s}Y\right)\right|
= 1, \] and prove that it has only finitely many solutions $(x,y, (t_1, \dots,
t_s) )$ with $xy \neq 0$ and $\mathbb{Q}\left( \gamma_1^{t_1}\cdots
\gamma_s^{t_s} \right) = K$, all of which are effectively computable.
