Rhys is polite, sincere, funny and easy to talk to. He's obviously interested in other people – I realise with horror when I play back my tape that I have spent nearly half the time wittering on about my job, my flat, my degree, my travels. His answers to my questions are lengthy and intelligent and thoughtful, but I get the impression he's just as happy quizzing me than talking about himself.
Rhys has played Kevin Walker in Brothers & Sisters for five years. That's a long spell in a profession where insecurity is a given. "It's like having a weird office job," he says. "I go to the same place every day, I park in the same spot, I turn up in shorts and flip flops and then put on a suit." His character – who happens to be gay, rather than being "a gay character" – is rarely without a shirt and tie.
Does he enjoy the routine? "It's nice to have security, but part of me misses the variety that comes with not knowing what you're going to do next." The cast (Sally Field, Rob Lowe, Calista Flockhart and Rachel Griffiths among them) are great mates – he was recently best man at his co-star Dave Annable's wedding.
Rhys's latest role is less stellar: he is bringing to wider attention, with first a documentary and now a travelogue, the extraordinary heroics of a small group of Welshmen in Patagonia, Argentina. Almost 125 years ago to the day, a party of 29 men left the safety of a small Welsh colony, established 20 years earlier, in search of more fertile land – and with it, survival – in the Andes. Their gruelling 700km journey, on horseback, took them five-and-a-half weeks.
To commemorate the 125th anniversary of the trip, descendants of the original men decided to recreate the entire journey. And Rhys persuaded them to let him come along.
"In my ignorance or arrogance, I underestimated the physical challenge of it," he says. "I didn't realise we would be in the saddle so long, and by the third day, I wasn't sure if I was up to it."
Part of the attraction was to document some real Welsh heroes "who make you feel good about being Welsh", he says. "I was a big cowboy fan when I was growing up. Wales doesn't have those kinds of heroes, so I wanted to get their story out."
He came across it when he stumbled on the diary of John Murray Thomas, one of the group's leaders. “He was a great adventurer. What they did was the equivalent of space travel, unknown and incredibly dangerous. We’re [the Welsh] not really known for our pioneer spirit, but here was someone who very much was.”
Rhys found extreme contentment on the month-long ride. "It was an immediately gratifying, real experience," he says. "You're not pretending to be anyone else."
Patagonia is the subject of his next film, out in March, in which he plays an Argentinean of Welsh descent.
He is drawn to stories like this. A few years ago he made a documentary about another extraordinary character, Griffith Jenkins Griffith, a Welsh-born industrialist who donated parks and public spaces to Los Angeles, but become better known for shooting his wife and spending two years in prison.
As well as promoting charismatic Welshmen, it seems the attraction of telling these stories is having a bit more creative control. "If you break up the elements of film acting, your input and satisfaction can be weak," he says. "You're reciting someone else's words, under someone else's vision, wearing clothes someone else has told you to wear. And the director and editor get to decide how it all turns out.
"I've started directing a bit on the show" – he has three episodes under his belt – "and I find it really gratifying. Everyone asks you how you'd like something shot, and you think, wow – I have an opinion on how this looks, what happens, the tone."
Rhys is about to reprieve, briefly, his role as Benjamin in The Graduate (he starred opposite Kathleen Turner on stage in London), this time for radio in LA. "I miss the theatre enormously. The Graduate feels like a lifetime ago but it was one of my best jobs. Everything clicked. The cast got on, we had a great social life with it, and it was great to make people laugh every night.
"It's weird how your perspective changes. At the start of your career you think, I just want to do cutting edge work that makes people think. Now, I would do a blockbuster in a heartbeat." What changes? "Just the awareness of what we do," he says. "Don't get me wrong, I take it very seriously and work very hard. But at the end of the day, we're entertainers."
Crossing the Plain, published by Gomer Press, is out now.
Black and white photographs: Matthew Rhys