TL;DR: The rule is each is always singular (apart from contrived contexts). But lots of learners and even some native speakers can be thrown by each of the [plural noun], and treat it as plural.
From grammar.ccc.commnet...
Each is often followed by a prepositional phrase ending in a plural word (Each of the cars), thus confusing the verb choice. [italics mine]
I expect most people would simply take it for granted that each is always singular, regardless of whether it happens to be just one word in the construction each of the [plural noun], but this NGram is interesting (it suggests the usage was much more common a couple of centuries ago).
And let's not forget indisputably valid usages such as We each of us have our faults. It takes some doing to figure out exactly why the presence of the apparently optional we there precludes us from continuing with the equally valid (as a complete sentence) Each of us has his faults (I'm guessing it's because the actual subject is plural we and that each is an adjective in an adjectival modifier, but what do I know?).
First, your impression:
The auxillary verb "to do" is used and must be placed in the short answer, as far as the rules demand.
I don't think you understand the definition of a "short answer". Let's look at a different question, where both British and American English have the same answers.
Did you take any candy?
Some of the possible answers are:
Yes, I took some.
Yes, I did.
No, I didn't take any.
No, I didn't.
But not
*Yes, I did some.
*Yes, I took.
*No, I didn't any.
*No, I didn't take.
The explanation is that "yes, I did some" doesn't count as a "short answer" here, while "yes, I took" counts as a "short answer", and so is ungrammatical because you didn't use the verb "do".
I am not going to explain all the grammatical rules about short answers—they're too complicated and there's probably another answer on this site which does it.
For your question, just replace the verb take with the verb have above.
Do you have any sugar?
Yes, I have some.
Yes, I do.
No, I don't have any.
No, I don't.
But not
*Yes, I do some.
*Yes, I have.
*No, I don't any.
*No, I don't have.
Best Answer
Given that it is a necessity that both instruments sound true during the show, we could say that the sentence is trying to express the urgency of tuning the instruments. Therefore, it can be that "have to be" is most appropriate in this context, although both versions are grammatically acceptable. As Dan added, context is lacking to precisely pin down the correct one.
Looking at difference in meaning between have to be and are to be:
have to be expresses a prerequisite for the show. If the instruments are not tuned, the show can't start. It is therefore closely related to "the instruments must be tuned" (but not the same)
are to be expresses a lower sense of necessity that the instruments are tuned, and rather indicates that doing so is general procedure. It has less emphasis on the "must", and instead it seems to convey that "this is always the case so the instruments will also be tuned this time". Like Grizzly suggested, it is expressing the expectation that the instruments are tuned.