In a statement, which was released to the PA news agency, the couple have said they have always tried to show the reality of life on the farm.However, they said the "constant intrusion" from some sections of the media has amplified a "rocky patch" the couple are going through.
It said: “With the TV show and the books we’ve always aimed to show the reality of life on the farm, and just like any marriage we have our stresses and strains, coupled with all the complexities of what we do on the farm and bringing up nine kids.
“We’re a normal family and we’ve never said our marriage is perfect. Unfortunately the constant intrusion into our lives from the media has amplified a rocky patch that we’re going through.
“We ask that the media respect our privacy as we work through this."
The couple have nine children, and have been married for 21 years.
The pair met in 1996, when Amanda was 21. Then working as a contract shepherd, she was sent to collect a ram from Clive's tenanted sheep farm, Ravenseat near Keld in Swaledale, and love blossomed. Amanda described the cottage as rundown with damp carpets and smoke-stained wallpaper, and she compared it to a TV programme called The Dale That Died.
Amanda has previously said she was more concerned with looking for a sheepdog than a boyfriend when she met Clive, who is also from a non-farming background and had never been to London. He had been living on the farm, which dates back to the Viking period, since 1989, moving from a farm in Stainmore which he ran alone.
He chose Ravenseat because of its position in the heart of Swaledale and associations with the breed of the same name.
In an interview with The Telegraph, she spoke of how Clive was so concerned with his flock that he survived on pies and cornflakes, and used one of his living rooms to keep feed bins in.
She says they didn't plan such a large family, but that each addition to their brood fits in well with their free-range life on the farm.