Jeremy Corbyn has spoken out over party discipline to the shadow cabinet after the shadow defence secretary Maria Eagle appeared to back the head of the armed forces when the general criticised the Labour leader’s position on Trident.

According to several party sources, Corbyn made it clear that disputes must not be carried out so publicly in future, as he addressed his weekly shadow cabinet meeting.

The Labour leader also reminded his colleagues that media statements must be cleared through his office and asserted that there must be stronger collective responsibility. It is understood Eagle’s remarks appearing to back the stance of Gen Sir Nicholas Houghton went down particularly badly with the leader’s office.

Houghton had suggested on the BBC’s Andrew Marr Show that he was worried about Corbyn gaining power because of the Labour leader’s opposition to renewing or ever using the Trident nuclear deterrent.

Eagle, who backs the renewal of Trident and is leading Labour’s review of the policy, then told the same programme: “I understand the point that [Houghton] is making. It is the point that I made myself when Jeremy said what he said.”

At the shadow cabinet meeting, Corbyn, who is writing to the Ministry of Defence to complain about Houghton, stressed his belief that the comments breached the principle of political neutrality and that he expected to be backed on that subject.

Some of those present said it was clear that Corbyn was warning that he would not tolerate any more comments that challenged his authority as leader. One said it was clear that it was meant as a rebuke to Eagle in particular. It is understood Corbyn’s comments were not at the time met with any objection by shadow cabinet ministers.

But some shadow cabinet members have told the Guardian that they are not happy about having to show Corbyn such discipline when they believe he has strayed from positions on issues such as remaining open-minded about Trident, as well as the fact he was one of the most rebellious MPs in parliament as a backbencher.

Trident is a particularly difficult subject for Labour because Corbyn was elected on a mandate of opposing renewal of the nuclear deterrent, but the party’s national policy currently supports it, as do many in his shadow cabinet including Eagle, Tom Watson, the deputy leader, and Hilary Benn, the shadow foreign secretary.

Scottish Labour, with Corbyn’s encouragement, has passed a motion opposing the renewal of Trident.

It is the issue which has caused the greatest anxiety over discipline, with a number of senior MPs directly and publicly opposing Corbyn.

Kevan Jones, shadow defence minister, recently wrote in the Telegraph strongly asserting his support for Trident and told The House magazine that he had worries about the danger that Labour would not be thought credible on defence.

Separately, Corbyn is expected to be sworn in as a member of the privy council on Wednesday, enabling him to receive confidential security briefings. He has confirmed that he intends to join the body, but has not said whether he will kneel on a footstool or kiss the Queen’s hand as part of the process.

Labour declined to give more details about how Corbyn will conduct the swearing-in ceremony. His choices may never be made public, because meetings of the organisation take place in private.

According to the Royal Encyclopaedia, the protocol is that: “The new privy counsellor or minister will extend his or her right hand, palm upwards, and, taking the Queen’s hand lightly, will kiss it with no more than a touch of the lips.”

The late former cabinet minister and leftwinger Tony Benn revealed in his diaries that he kissed his own thumb instead of the Queen’s hand.

Some Tory MPs and the Labour MP Simon Danczuk criticised Corbyn for missing the first meeting because of prior engagements.

The issue of Corbyn’s membership has been much discussed because he is a lifelong republican. Michael Foot and Neil Kinnock both became privy counsellors, which entitled them to higher-class security information and briefings.

