Using postings limited to 140 characters or less, the British-based actors have been asked to respond to each other, to the Twitter "audience" and to real events as they happen around the world.
The actors' 'dialogue' will be relayed to computers from their mobile phones.
Juliet, played by 19-year-old RSC actress Charlotte Wakefield, is a teenager who has never had a boyfriend, but finds solace by chatting incessantly on the Internet.
For the purposes of the story, her mother, Susan Capulet, 34, was killed in a car crash a decade ago.
In one early tweet Juliet linked a YouTube video she had made of her room, pausing on the framed photo of her dead mother.
On her way to school she tweeted, complete with grammatical errors: "Ok now dads beeping at me from the car!! Gotta dash!! Wish I could tweet at school but... I can't :( I promise I'll be back on after school.
"Jules is over and out!! Xxx"
A typical tweet from Juliet's brother Tybalt, who risks being expelled from school because of bad behaviour, was: "Gonna be late for class coz i've gotta have breakfast. Couldn't give a crap!"
The actors have been asked to improvise around a prepared story "grid" set in modern Britain, rather than the Verona of the original, to ensure events happen at a certain time.
They are writing the tweets themselves, taking inspiration from their character backgrounds and a detailed diary that has been given to them.
The project, named "Such Tweet Sorrow" in a play on the words "parting is such sweet sorrow" from 'Romeo and Juliet', is a co-production with a company that produces entertainment on mobile phones, TV and the Internet.
RSC artistic director Michael Boyd said the world-renowned theatre company's aim was always "to bring actors and audiences closer together".
"We look forward to seeing how people engage with this new way of playing," he said.
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