The Royal Family are "hurt" at the Duke and Duchess of Sussex's announcement they will step back as senior royals.
Prince Harry and Meghan did not consult any royal about making their personal statement, BBC royal correspondent Jonny Dymond was told by palace sources.
Buckingham Palace was "blindsided", he said, as talks about their future had begun but were in very early stages.
The palace said there were "complicated issues" to work through.
In their statement on Wednesday, Harry and Meghan said they made the decision "after many months of reflection and internal discussions".
They said they intend "to step back as 'senior' members of the Royal Family, and work to become financially independent".
They plan to split their time between the UK and North America, while "continuing to honour our duty to the Queen, the Commonwealth, and our patronages".
"This geographic balance will enable us to raise our son with an appreciation for the royal tradition into which he was born, while also providing our family with the space to focus on the next chapter," the couple said.
Despite the couple's decision, Harry will remain sixth in line to the throne.
The pair was already preparing to launch their own Sussex Royal charity, which they set up after splitting from the Duke and Duchess of Cambridge's foundation in June last year.
The Sussexes' new charity is expected to be global, linked to Africa and the US, rather than domestic - and will have a commitment to female empowerment.
It was revealed in December the couple had made an application to trademark their Sussex Royal brand across a string of items including books, calendars, clothing, charitable fundraising, education and social care services.
Why North America?
Meghan, of course, is American herself and has a particularly close relationship with her mother who lives in California.
But the Sun, which broke the story on Wednesday, was firm in its speculation that Canada was their likely destination.
The couple have just returned after a six-week break from royal duties, which they spent in Canada with their eight-month-old son, Archie.
After returning to the UK on Tuesday, Harry, 35, and Meghan, 38, visited Canada's High Commission in London to thank the country for hosting them and said the warmth and hospitality they received was "unbelievable".
Former actress Meghan lived and worked in Toronto for seven years during her time starring in the popular US legal drama Suits.
She has previously spoken about adopting the city as her second home after her native Los Angeles, and she has several Canadian friends.
Jessica Mulroney, a close friend of the duchess, posted a quote on Instagram about a "strong woman" facing a challenge.
The Canadian stylist and TV star wrote: "A strong woman looks a challenge in the eye and gives it a wink."
The couple's announcement raises numerous questions over what their future royal roles will entail.
How will they become 'financially independent'?
In stepping back as senior royals, Harry and Meghan have announced they will no longer receive funding from the Sovereign Grant.
The couple said this will make them "members of the Royal Family with financial independence".
The Sovereign Grant is public money which pays for the cost of official royal duties, in exchange for the surrender by the Queen of the revenue from the Crown Estate.
The Queen's Sovereign Grant from the Treasury was £82m in 2018-19.
The couple said the Sovereign Grant paid for 5% of their official office from 2019, with the remaining 95% being funded by Prince Charles through his income from the Duchy of Cornwall.
Under current rules, the pair said they are "prohibited from earning income in any form", but in their new roles could follow other title-holding royals in having full-time jobs.
The costs of official overseas visits will be funded by the Sovereign Grant and contributions from the host country "when appropriate".
The couple are classified as "internationally protected people", which means they must have armed security provided by the Metropolitan Police.
They will retain Frogmore cottage, the Grade-2 listed property in Windsor that cost taxpayers £2.4m to renovate, as their official residence so they have a "place to call home" in the UK.
What do commentators say?
Bryony Gordon, a Daily Telegraph journalist who knows and has interviewed the couple, suggested their decision could be linked to their mental health, after becoming a "punching bag" for a "misogynistic and racist" nation.
In 2017, Ms Gordon interviewed the prince for her Mad World podcast in which he revealed he sought counselling while struggling in his late 20s to cope with the death of his mother.
Speaking to Emma Barnett on BBC 5 Live, she said: "Here is a man who every day has to live out the trauma that he experienced as a little boy when he had to walk behind her coffin at the age of 12 in front of the world, and I think if any of us were put in that situation we would find it incredibly triggering."
She added the couple had become a "punching bag for a nation that is still sort of deeply misogynistic and racist" and they were "entitled" to "do what is best" for their family's mental health.
She believes the couple will carry on exactly as before "but just for free", as they're "sick of the scrutiny" that comes with taking public money.
Royal historian Prof Kate Williams said it will be "difficult" for Harry and Meghan to lead "normal" lives, as media interest in them will increase after their "unprecedented" move.
She told BBC Breakfast: "It is going to be difficult if Harry and Meghan are going to live in Canada for a certain period of time and try and get on with business and be normal, charitable CEOs.
"They will require security because I don't see the media interest in them waning, I see it as probably increasing, because what they're doing is so unprecedented for royals."
Royal commentator Penny Junor said the couple's actions were reminiscent of those of Harry's mother, Diana, Princess of Wales. The suggestion that they had not consulted other royals before making their statement was "beyond bizarre".
She said: "It has echoes of Diana when she suddenly announced after her separation [from Charles] that she was stepping back from 50 of her charities without consulting anybody.
"The problem is that they are not working for themselves, they are working for a family firm and to be making announcements of this sort without consulting is beyond bizarre."
Graham Smith, a spokesman for Republic, which campaigns for an elected head of state, said Harry and Meghan's decision "raises questions about the monarchy's future" and will prompt taxpayers to ask how the couple's extra security and overseas lifestyle will be funded.
"To suggest that they're not already financially independent is incredibly crass and belies a sense of self-entitlement and a lack of self-awareness that is common among royals," he said.
"This really is wanting to have your cake and eat it. They have said they will dip in and out of royal duties as it suits them but won't stop taking public money until they find other sources of income."
Justice Secretary Robert Buckland said "we must continue to celebrate and support" the contribution the Royal Family makes to the UK, adding that it has an "important role in public service".
A staging post to leaving the Royal Family?
This is clearly a major rift between Harry and Meghan on one part, and the rest of the Royal Family on the other.
There are far more questions than answers. What will their new role be? Where will they live, and who will pay for it? What relationship will they have with the rest of the Royal Family?
And there's the institutional question - what does this mean for the Royal Family?
It comes just months after Prince Andrew stepped back from his duties after a BBC interview about his ties to sex offender Jeffrey Epstein.
Some might see this as the slimmed-down monarchy that the 21st Century needs.
But Harry and Meghan reached people that other royals didn't. They were part of the reinvention and refreshing of the institution.
It's understood talks have begun between Palace officials and No 10 about the implications of Harry and Meghan's announcement.
We're now in wait-and-see mode as to whether this new model of being a royal can work - or if this is really a staging post for them to leave the Royal Family.
Was it a surprise?
Signs the couple were unhappy with their royal life have been apparent for some months.
In October last year, the Duchess of Sussex began legal action against the Mail on Sunday over a claim that it unlawfully published one of her private letters. The paper stands by its story.
At the time, Prince Harry said his wife had "become one of the latest victims of a British tabloid press" after a "ruthless campaign" of "relentless propaganda".
"Though we have continued to put on a brave face... I cannot begin to describe how painful it has been," he said.
In an ITV documentary last year, Meghan, who was born in the US, described motherhood as a "struggle" due to intense interest from newspapers. "Not many people have asked if I'm OK," she said.
Shadow health secretary Jonathan Ashworth said on Thursday the press should "leave Harry and Meghan alone" in the wake of their decision.
The Labour MP for Leicester South expressed support for the couple, adding that the duchess had suffered "appalling" abuse and that she "deserves a break".
A new media strategy
On Wednesday the royal couple announced they would be adopting a "revised media approach" from the spring.
As part of the new strategy, published on their website, they will "engage with grassroots media organisations and young, up-and-coming journalists".
They will also pull out of the so-called royal rota system, where journalists and media representatives are given exclusive access to cover royal engagements on the understanding they share the material they gather.
"The current system predates the dramatic transformation of news reporting in the digital age," the couple said.
The National Union of Journalists criticised the plans, saying the move appears to try and "prevent the media from functioning and compromising the ability of journalists to do their jobs, which is completely unacceptable".
Harry is sixth in line to the throne - behind Prince Charles, Prince William and his three children.