There is a difference between "real" and "based on reality". The Romeo and Juliet legends are (loosely) based on reality. Shakespeare's play was inspired in part by an epic poem by Albert Brooke called "The Tragical Historie of Romeus and Juliet", which had been written decades earlier. Brooke in turn was inspired by an Italian novella written years earlier by Matteo Bandello. Bandello was rewriting a story from Luigi Da Porta. They and all the other "Romeo and Juliet" authors were writing about events that occurred during the very early 1300s in Verona, where a civil war of sorts was in progress between two political factions, the Guelphs and the Ghibellines.
The facts: Verona was ruled by Prince Bartolomeo I della Scala, who really was trying to keep peace between the factions. And Dante really was living in Verona at the time as a guest of the Prince. The names "Capulet" and "Montague" are mentioned cryptically in his "Purgatorio". He says they are "grieving", but does not mention why.
If the fact list seems short, that's because it is. :D
The LEGEND is that a girl from one faction killed herself after her lover from the other faction was killed in the fighting. Everyone felt so sorry for her that she was buried in consecrated ground under the church, even though she was a suicide. There is no Nurse, no scheming priest, no marriage (arranged or otherwise), no Mercutio, no sleeping potion---all of that is grafted onto the story later. In some early versions of this story, their names are not Romeo and Juliet. In other versions, Romeo is hanged.
The Juliet house in Verona is an example of what my husband likes to call "un trappo di tourista". They've been capitalizing on the success of Shakepeare's play there for centuries.