A 39-second video posted on their official Twitter account shows the couple climbing up sand dunes and enjoying time in the countryside with their family.
In a message accompanying the video, the couple, thanked those who had sent “kind messages”, adding: “We are enormously grateful for the 10 years of support we have received in our lives as a family.”
In her role as a senior royal Kate has focused her charity work on early years, young people’s mental health, the visual arts and promoting the benefits of outdoor life.
During the Covid-19 crisis, the Cambridges have been supporting key workers and frontline NHS staff with video calls and face-to-face engagements when permitted, and promoting the vaccine drive.
They also became joint patrons of NHS Charities Together.
The traditional gift on a 10th wedding anniversary is tin, symbolising strength and durability.
The Cambridges have faced personal challenges over the past year, dealing with the fallout from Megxit, William’s rift with his brother, and the Sussexes’ bombshell Oprah Winfrey interview.
The Sussexes plunged the monarchy into crisis weeks ago when they accused the royal family of racism and the institution of failing to support Meghan when she was suicidal.
Kate was publicly singled out by Meghan as having made her cry in the run-up to her 2018 wedding in a disagreement over flower girl dresses.
The Cambridges’ anniversary is a moment of joy during a period of great sorrow for the royal family, who are grieving for the Duke of Edinburgh, who died earlier this month at the age of 99.