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This thread is resolved. Here is a description of the problem and solution.
Problem: Client is asking "Can I change the theme on our website without any problems on our multilingual site?"
Solution: First of all, it is always a good idea to create a full website backup before making updates or changes.
Regarding the question, there should not be any issues when you change the theme, however, what will happen is that you might need to translate new strings that come with the new theme.
If the content that you translated already was not dependent somehow on the theme that you were using, then it should remain unaltered.