Let $M$, $N$ be two countable structures in an uncountable language $L$. Suppose that for any countable sub-language $L'$ of $L$, $M$ and $N$ are isomorphic $L'$-structures. Then, can $M$ and $N$ be non-isomorphic $L$-structures?
I found this problem when I thought about Does any uncountable complete theory have exactly two countable models?. Understanding restrictions to countable languages might help to understand uncountable languages.
Any help would be appreciated. Thank you.