Shaw rejected romanticism and embraced realism. How realistic is Arms and the Man? How much of it is "unrealistic"?

This dichotomy really oversimplifies and misrepresents Shaw. Shaw was first and foremost a satirist whose plays tend to debunk popular misperceptions using humor and irony. Thus Arms and the Man is anti-Romantic in terms of its views of war, love, and a certain tradition of war literature, but it achieves its effects through elements of hyperbole, stereotyping, and coincidences, albeit often with surprising twists. 

For example, Shaw debunks the notion of the "war hero" in...

This dichotomy really oversimplifies and misrepresents Shaw. Shaw was first and foremost a satirist whose plays tend to debunk popular misperceptions using humor and irony. Thus Arms and the Man is anti-Romantic in terms of its views of war, love, and a certain tradition of war literature, but it achieves its effects through elements of hyperbole, stereotyping, and coincidences, albeit often with surprising twists. 

For example, Shaw debunks the notion of the "war hero" in his portrait of Sergius, but the plot twist of the incorrect ammunition is highly improbable. Similarly, an "anti-Romantic" point is argued with Raina's choice of a middle aged Swiss mercenary over the glamorous, handsome young Serb; I am not sure that a real teenage girl would make that choice (at least before discovering the wealth of the Swiss soldier). 

Shaw himself admits that he uses much of the scaffolding of the pièce bien faite in creating plots, rather than using a realistic plot structure. The plot of the play is that of a romance in which the two pairs of lovers first are attracted to the wrong people and eventually marry the right people, a structure that is quite conventional and not realistic.