Lost
I do not consider this "romance" ... The relationship between the leads cannot be so easily defined. Whatever romantic undertones are so, so subtle. For most of this drama, the characters are not ready for romance, and the story does not force it. Their relationship is primarily a very deep, healing friendship. It's about finding solace in the company of a stranger.
Really, this drama is just so melancholy. It's beautiful and sad, slightly depressing but also uplifting. This drama truly knows how to use silence to the best effect. Nothing could make me not recommend this enough. It fills a hole in your heart that you didn't know you needed. Please watch it, really.