Che is Argentina itselfNarrator/everyman that appeals to Eva's conscience to help the poor like she promised? Works with the people, particularly during A New Argentina? Makes sense that he'd have a vested interest in the country if he IS the country. Also, who better to narrate a story about Argentina's most famous citizen than the country himself?
At least in the film version, Che is a projection of part of Eva's personalityThis explains why he's always around Eva, having apparently followed her from her hometown to Buenos Aires. It also explains why she doesn't notice that her neighbor, some guy from her hometown, the barkeep, and roughly half the waiters all look the same.
Che is a Time LordHe got stranded on Earth and decided to amuse himself by following the life of one Earthling who caught his interest. The reason she never realised that he was following her is because he was using a low level perception filter to stop people from noticing. The real clincher however is the waltz for Eva and Che, for it was established in 'The Girl in the Fireplace' that Time Lords can interact with people inside their heads. He mentions that he might "seek worthier pastures, and thereby restore self-esteem", aka, he's grown tired of following her.