Kaley Cuoco and Henry Cavill
07/04/2013 at 11:00 AM EDT
It's a bird, it's a plane ... it's Superman's kryptonite, Kaley Cuoco
Days after a hike through the hills
followed by a sushi stop-off, Henry Cavill, 30, and The Big Bang Theory
actress, 27, snuck in a little PDA while getting groceries in Los Angeles.
The actor – true to Man of Steel
's signature colors in a red tee and jeans – kept a good grip on his real-life Lois Lane, who stayed well shaded in a brown fedora, matching sunglasses and a leopard-print top.
After stocking up on goodies, the two later emerged from the store with big smiles, a cart full of brown bags – and a trail of adoring young fans.
"It's just the beginning stages of a relationship and they're having a great time," a source recently told PEOPLE