For two distinct prime numbers $l$ and $l’$ let $V_l$ (resp. $V_{l’}$) be finite-dimensional vector spaces over $mathbb{Q}_l$ (resp. $mathbb{Q}_{l’}$).

Suppose when regarded as $mathbb{Q}$-vector spaces $V_l cong V_{l’}$.

Is there always a $mathbb{Q}$-vector space $V$ such $V_l cong V otimes mathbb{Q}_l$ and $V_{l’} cong V otimes mathbb{Q}_{l’}$?

Does the answer generalize to the case of not just two but infinitely many primes?