This is a strange land, all right - and I think we're going to be in hot water over the wording of this puzzle, Warren. But here goes:
The first person must have said, "I am a man", because if he were a man, he'd tell the truth and say so; and if she were a woman, she'd lie and say she was a man. So the second person's first statement was true - which means the second person is a man. And because the second person is a man, his second statement must be true too - which means the first person is a woman. Also because the second person is a man, the third person's first statement is false - which means the third person is a woman. The second person's third statement and the third person's second statement are both superfluous, but you know how natives love to confound naive explorers who ask silly questions like, "Say there, are you a man or a woman?".
The provided answer to this conundrum is not entirely correct. You see, the entire logical reasoning is grounded on the premise that the first person said " I am a man." This premise is based upon the presumption that, no matter the sex of the first person, this would be a possible answer to the question. If the first person was a man, he could enuciate this truthful expression of his gender, and if she was a woman, she could also provide this same expression, although flase, based upon the provided algorythm.
Nevertheless, it is a fallacious assumption to presume that the first person did in fact say these words. It is entirely possible that the first person would have replied to the interrogative with the words "I am not a woman," or even perhaps a more indignant response.
In this event, we can no longer conclude based upon the first statement of the second person that the second person is a man. Furthermore, this case leads to a breakdown of all of the conclusions arrived at in the provided discourse. Indeed, all we can conclude no is that the first and third persons are of the smae gender, and that the second person is of an opposite gender.
I hope this commentary has been enlightening.