For a phrase such as the following:
each apple and each orange
Is it correct to use "has" or "have" when describing properties of both apples and oranges?
Both each apple and each orange are respectively singular so it has to be has. If apples and oranges (plural) were the subject, you could use have.
Let's change each to reveal its meaning, and see what happens.
We can now more easily see that each refers to the singular nature of one apple, and the singular nature of one orange. But more than that, it emphasizes that each of these things stands alone, and functions independently as the subject of the sentence. By using each in this way, we prevent the apple and the orange from being linked into a group; the use of each in fact separates them. Therefore, the verb remains singular, and you would say "has," not "have."
Using each in this way is equivalent to separating the two things and constructing an independent clause which we then use for each one, such as in this example:
Note that I have changed each to an. I can do this because repeating the independent clause makes it clear that I am referring to only one thing at a time (or one class of things; "an apple" sometimes refers to apples in general). And quite naturally, I use has for each clause. We don't usually use this kind of repetition, though, because it is awkward, and that is why the use of each is effective, thus: Each apple and each orange has a stem. That's more succinct and efficient.
Also note that an isn't the only word that can be used in this place; it's just an example. I could have said, "That (or my or one etc.) apple has a stem, and that (same options) orange has a stem." And I could even have used each, thus: "Each apple has a stem, and each orange has a stem." In this case, however, the use of each is different; here it actually means I am talking about every one of the whole group of all the apples. (English allows for lots of subtly different usages of the same word. That's what makes this web site so interesting.)
On the other hand, if I had said "an apple and an orange," (keeping the two things together, and not separating them by sticking a clause in between) this would link the two objects into a group, and would create a plural subject for the sentence. You would then say "an apple and an orange have stems." (This would still be referring to apples an oranges as classes of objects, but now the classes are joined into a plural subject.)
So, just to reiterate, in the case of your example, "has" is the correct choice.
6/19/13 ADDENDUM TO RESPOND TO TCHRIST'S QUESTION:
Each does in fact change a coinjoined subject into two (or however many you are referencing) singles. But asking why it does so seems to me to be similar to asking why a car transports people from place to place. It does so because that's exactly what it is supposed to do. That is its function. Also, and interestingly, your examples of "both pieces the same" do not quite seem the same to me. "The master and commander" refers to one and the same person, does it not? And yet even in this case, you could put each in place, and suddenly you would be referring to two different people. Talk about a separation! In the second example, "time and tide" are two separate entities, and you could separate them with each if you chose, but of course I needn't tell you that that would destroy the well known axiom, changing both the poetry and the meaning in a way that wouldn't suit what you are trying to say. And yet, it would be valid syntactically, and would emphasize the oneness of each of these entities.
It is clear that apples and oranges are completely different fruits, so we are not talking about a specific group of students, political candidates, houses, or countries etc. However, these two fruits do share one quality.
It is the conjunction "and" which tells us the apples and oranges share this one feature. It is the adjective, "each", that explicitly tells us which apples and oranges; every single one. Consequently, the verb “have” agrees with the individual fruit (singular subject) thus explaining why "has" and not have is preferred.
If the author wishes to use the plural verb, have, for stylistic reasons, nothing forbids him from doing so. He merely needs to change the wording and reposition the small but very useful word, each in the sentence.