As generally agreed and as extensively discussed in this question, "an" should be used in place of the more common "a" where the following word begins with a vowel sound.
I have just encountered for the first time an instance of the phrasing
in the book Why Chemical Reactions Happen (James Keeler and Peter Wothers, Oxford University Press, 2003). (This book is not mine. I was sitting opposite a friend when the friend mentioned the unusual phrasing.)
This phrasing appears to contradict the rule discussed above, since "mule" is, as far as I am aware, pronounced /mjuːl/ (see Wiktionary).
Here is the context. The author is using an analogy to explain a chemistry concept (the italics are not mine, but the bold emphasis is):
A useful analogy here is that of an mule, which we get by crossing a horse with a donkey. To describe an mule as a horse or a donkey would be inaccurate and it certainly does not interconvert rapidly between the two! Rather we need to recognize that although an mule has parts that are reminiscent of a horse and parts that a reminiscent of a donkey, it is something altogether different.
Looking for other examples online, I have managed to find one on Wikipedia's List of Siege engines. Discussing a Siege engine, the Scorpio:
Similar to the ballista, but smaller. Was sometimes mounted on an mule-drawn cart.
Other examples are difficult to find, but here's one from a user-posted question on Answers.com (referring, notably, to a kind of deer rather than the animal, a mule):
Is an mule deer an omnivore?
No. Mule deer are herbivores. ...
What is going on? Is there an irregular pronunication of "mule" of which I am not aware? Is this some strange language quirk, some archaic spelling that I am encountering for the first time?