Henry Cavill and Kaley Cuoco Step Out – Holding Hands

Henry Cavill and Kaley Cuoco Step Out – Holding Hands
Kaley Cuoco and Henry Cavill

07/04/2013 AT 12:00 PM 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.

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