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Minghella was born in Hampstead, London, the son of director Anthony Minghella (1954–2008) and dancer and choreographer Carolyn Jane Choa. His father was born in Ryde, Isle of Wight, and was of Italian descent, and his mother was born in Hong Kong. His maternal grandfather was of three-quarters Han Chinese and one-quarter Jewish descent, and his maternal grandmother was of Chinese, English, Irish, Parsi, and Swedish ancestry. His maternal great-grandfather was Hong Kong legislator Robert Kotewall, and his maternal great-great-grandfather was Calcutta-born businessman Emanuel Raphael Belilios.
Minghella spent time on his father's film sets while growing up, of which he has "fond memories" and has said that he felt "no pressure" by his father to succeed in the entertainment industry. He was educated at St Anthony's Preparatory School and University College School in Hampstead. He then attended Columbia University, which he considered his "first priority", and studied history, graduating in 2009. He typically worked on films only during his summer break. He said that he felt like "an English boy at an American school", that he kept to himself, and that most of his fellow students did not know that he was an actor.