nt.number theory – Base change for vector spaces

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?