This is the first high-profile relationship for Tilda, 22, who has been dubbed the 'Aussie actress you need to know' by Marie Claire magazine.
The Adelaide-born rising star first became recognisable in Australia after she starred in Myer's Find Wonderful commercial in 2014.
The budding actress - who is about to appear in new film Hotel Mumbai with the British star - teamed her statement coat with a red and white striped shirt.
The British man of the moment, 26, looked handsome and carefree as he wandered arm in arm with his new love.
The BAFTA winning actor had clearly made an effort with his appearance for the date.
She also showed off her long slim pins in dark denim jeans and, sharing a giggle, the two looked adoringly at one another as their hands intertwined while they walked.
After hitting up a gastropub in East Hollywood for lunch, the lovers strolled down the road and checking out some local book stores.