The 27-year-old has also revealed that he was once asked to sign a fan's packet of Jammie Dodgers, reports The Sun.
"It's weird," he said. "Even going to Tesco is a different experience now.
"I was in a hotel and someone got out a pack of Jammie Dodgers and said, 'Could you sign these, please?'"
"But it's fine," he continued. "You try to give people some time. And then you run.
"It is overwhelming. But you just have to deal with it. It was quite unexpected, out in the US."
He added: "Fans in the UK are just as passionate, but they are more reserved, I think."
Smith is currently filming the Doctor Who Christmas special.
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