April 28, 2014 12:00 PM

Yes, Emma Stone and Andrew Garfield are that couple. The ones who hold hands during dinner. The ones who use endearing pet names for each other. (She’s “Em”; he’s “Garfy.”) The ones who take surfing lessons together in Hawaii and hop on a Vespa together in Rome. In short, they’re the kind of couple whose earnest lovey-doveyness would be annoying if they weren’t so irresistibly cute. “They almost seem to be in their own world when they spend time together,” says an L.A. source. “They have the kind of loving chemistry that makes other people jealous.”

Plus they’re young, gorgeous, rich and just finished with an international press tour of the world’s most romantic cities – London! Paris! Rome! Is there any reason not to be jealous? And yet even as Stone, 25, and Garfield, 30, promote their roles as comic-book sweethearts Peter Parker and Gwen Stacy in The Amazing Spider-Man 2 (out May 2), they are also the kind of couple who shrug off attention on their real-life romance of nearly three years. Avoiding the nightlife scene in favor of cozy nights in with golden retriever Ren at Stone’s four-bedroom Beverly Hills home or dates at L.A.’s Sushi Park, the pair have resolved to take a guarded, if playful, stance on their relationship. Asked recently if he had costume approval for his skintight Spidey suit, Garfield replied, “I had one focus group, and that focus group was Emma. She approved.” But away from the spotlight, they are anything but coy. “It’s very comfortable between them,” says a source who was with them during their recent Paris press stop. “When Andrew talks about Emma, a light comes over him. He gets starry-eyed.”

Both stars have acknowledged that sparks flew when Stone first auditioned for Spider-Man in 2011 after a series of actresses had already tried out. “Em was the last, on the last day,” Garfield told USA Today in 2012. “I was knackered. I was toast. She came in, and I woke up.” In between scenes, he added, “she just made me laugh.” For her part, Stone said, “at the audition, I was instantly like, ‘I could learn so much from this person.’ ”

They grew up on different continents – she was raised in Scottsdale, Ariz.; he in Surrey, England – but both have said they felt like outsiders. As a kid, Stone suffered from panic attacks that left her “immobilized,” as she told Vogue in 2012. She found solace in acting and quickly showed off her comic timing in films like 2010’s Easy A and 2011’s Crazy, Stupid, Love. Garfield, who was born in L.A. but moved to England as a toddler, attended drama school at the University of London before breaking out in 2010’s The Social Network.

Both close to their families (who joined them on their Hawaiian vacation in January), the couple have shown a natural ease with their young fans throughout the Spider-Man press tour, including a visit by Garfield to Kids’ City in London, an after-school club for disadvantaged youth. “Andrew arrived as Spider-Man, carrying some pizza,” Kids’ City chief executive Jackie Nunns tells People. “The kids were completely dazzled.”

The couple next head back to the U.S., with Stone set to attend the EIF Revlon Run/Walk in New York City on May 3 in support of her mom, Krista, a breast cancer survivor. Is an engagement likely anytime soon? A source in their circle says no – and the pair appear in little rush. “Their relationship seems very real,” says the L.A. source, “and very sweet.”

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