David Stewart writes ....
It's easy to believe there would be some real antagonism there.
You can imagine the relationship got off to a rocky start. Cynthia Lennon, John's first wife, was a friend of the Beatles and had known them since before their Cavern Club days. Paul's former girlfriend was a housemate of Cyn's and they used to spend a lot of time together. Cyn was more than just their friend's wife, she was a member of the Beatle's inner circle and the mother of a boy they all felt fond of (Hey Jude was written by Paul for Julian). She had been part of their lives for 10 years when she came home early from a holiday and found Yoko in her house wearing her bathrobe. Yoko was the woman who broke up John's marriage and the two of them treated Cynthia appallingly from then on.
The relationship didn't develop well from there. For years prior to John's relationship with Yoko, the studios at Abbey Road were a sacred space. It was a place for John, Paul, George and Ringo to make their magic with George Martin. Even Brian Epstein talked about how he felt out of place when he entered. None of the other Beatle wives or girfriends spent any time in the studio but suddenly Yoko felt it was her right to be there (fully supported by John). She not only felt the need to be there, she felt she had a right to make musical contributions and even had a bed brought in at one point when she was feeling unwell. Imagine how you'd feel if your work colleague's spouse started turning up to your work place and giving their opinion on what you doing, despite having no aptitude or talent for what you were doing? Imagine the work place was rearranged to accommodate her? How would you feel?
After the Beatles break up there was a really ugly period of antagonism between Paul and John. They sniped at each other in song and in interviews and were generally unpleasant to each other whenever possible. Yoko was by John's side in this era agreeing with everything he did. While Paul and John were able to return to civil conversation in the years just before his death, Paul never had the history of friendship with Yoko to fall back on. It's easy to imagine he still harboured a lot of resentment for her.
In the years after John's death, Paul has had to deal with the frustrations of Yoko's ownership of John's legacy. John and Paul's partnership lasted for around 12 years and Yoko was only involved for the last few years of the Beatle's life but Paul still required Yoko's involvement in things that she had nothing to do with. When they were compiling the anthology series, Yoko had to be consulted about which tracks would be included. She could determine which tracks from the Decca audition, which occurred years before she ever met John, would be made public. That must have annoyed Paul a lot.
It's hard not to imagine Paul feels a large degree of resentment at the woman who broke up the marriage of his friends, caused problems in his workplace, was part of an extremely bitter feud and now claims equal authority over events that she had no part in.