Reported Speech with was and were

I set a test with the following direct sentence, which the students were supposed to put into reported speech: Interviewer: "Are you fluent in English?" I expected the students to write: The interviewer asked me if I was fluent in English. However, some students wrote: The interviewer asked me if I were fluent in English.

I know that "was" and "were" are sometimes both possible, like: If I was/were rich... But to my knowledge this is used in unreal situations. Is it correct to say: The interviewer asked me if I were fluent in English?

I'm hesitant to disagree with @Barrie-England given his credentials, but I don't believe "were" would be grammatical in this case. As he mentioned, "it conveys varying degrees of remoteness from factuality." In your example, it's unlikely that you're asking your reader about his or her wishes, opinion, judgement, or possible state. Your statement is asking for clarification of a definite and factual characteristic.

Answering the question with a "were,"

The interviewer asked me if I were fluent in English.

would leave the audience (me at least) wondering what the uncertain event, action, or state would be. Expanding it to something like the following to add the necessary element of uncertainty would seem grammatical to me.

The interviewer asked me if I were fluent in English, would I then travel to England.

That's probably an awful example, but I see "were" being used with words that denote uncertainty: would, if, should, could, wish, must, etc.

I see "was" being used to denote certainty. Presuming your question was expected to be answered with a "Yes" or a "No," "was" should be used.

Really, though, shouldn't they have responded with:

The interviewer asked me if I am fluent in English.

Both are grammatical. Were in contexts like this is usually called a subjunctive, but the authors of 'The Cambridge Grammar of the English Language' use the term ‘irrealis were’. They write that ‘it conveys varying degrees of remoteness from factuality’ and that were is more formal than was.

My own view is that if I was is the more natural form.