SALZBURG, Austria — On the most difficult Brexit problem, the two sides are as far apart as ever.

The acrimonious end on Thursday to an EU summit in Salzburg, Austria — at which EU leaders savaged Theresa May's Brexit plan — demonstrates just how big the gulf between London and Brussels still is on the key issues. According to senior officials and diplomats from both sides, who spoke on condition of anonymity, it is the conundrum of how to prevent a hard border in Ireland after Brexit that is proving most difficult to solve.

Despite seeming taken aback by the forceful rejection of her Brexit plan by her EU counterparts, May hopes a proposal for a special customs status for Britain beyond 2020 could provide a solution.

In order to try to find a way through, the EU is now considering writing a new clause into the final exit package — as part of the future relationship. If it makes the final agreement, it would, in effect, create a second U.K.-wide transition phase that would come into effect in 2021 and last for a set period of time, one senior EU27 official said.

“The possible need of a bridge will be spelled out,” the official said.

Varadkar said negotiators are no closer to reaching a Brexit divorce deal than they were in March.

Disagreements remain. The U.K. wants the “bridge” clause to be legally bulletproof, written into the withdrawal treaty, and not just a statement of political intent. The EU has, so far, shown no willingness to entertain this and it is not backing down from its demand that the Withdrawal Agreement include a legally binding guarantee of no hard border on the island of Ireland.

The Irish border chasm at the heart of the Brexit talks was on full display in Salzburg, which had been billed as an important opportunity for EU leaders to steer the Brexit talks as they enter their final weeks.

Irish Prime Minister Leo Varadkar said negotiators are no closer to reaching a Brexit divorce deal than they were in March. "We need to redouble our efforts over the next couple of weeks," he told reporters Thursday.

At a press conference closing the summit, European Council President Donald Tusk said: "There will be no Withdrawal Agreement without a solid, operational and legally binding Irish backstop."

Tusk was reaffirming Brussels' insistence that there be an ironclad "backstop" that can be relied on in perpetuity to prevent the need for a hard border in Northern Ireland, whatever else is agreed between the two sides. The EU27 say this must be a legal reality that is written into the Withdrawal Agreement and comes into force the day after the transition period ends on December 31, 2020. It applies unless and until something else can be agreed to take its place in the meantime.

"We need a U.K. proposal precisely preserving the backstop in the framework of a withdrawal agreement," said French President Emmanuel Macron.

London disagrees.

Although May signed up to a backstop in December last year, she strongly rejects the EU's conception of it. Brussels' demands, she has said, would pose a threat to the economic and constitutional integrity of the U.K. that "no British prime minister" could agree to.

Her position is that the “backstop” should not be a permanent legal reality, but a temporary “bridge” to whatever new trade and customs relationship both sides are able to negotiate, two U.K. officials said, speaking on condition of anonymity.

In effect, the U.K. is trying to turn the backstop into a second transition period. In London's view, it would be time-limited and apply to the whole of the U.K., not just Northern Ireland.

For the EU27, leaving the details of the backstop out of the withdrawal treaty would not provide enough legal certainty.

May has proposed that during this period, the U.K. will stay within the EU’s customs regime — if a long-term trading arrangement with the EU has not yet been agreed. London will release details of how close it intends to align with single market regulations during this bridge period in the coming weeks.

May says this would keep the border open in Ireland while also ensuring there is no customs barrier between Great Britain and Northern Ireland.

For Brussels, this amounts to cherry-picked EU membership for the whole of the U.K. — something it will not accept.

There's a second component to London's offer.

The U.K. says it is willing to accept a second clause, described in Whitehall as a “safety net,” that would protect the Irish border in the event London later decides to change whatever trade relationship it has negotiated with Brussels.

The problem for the EU is that the U.K. has proposed putting this into the “political declaration” laying out a joint vision for the future relationship — not the binding withdrawal treaty setting out the divorce terms.

“You can’t negotiate a permanent solution before a temporary solution,” said one senior U.K. official. “That doesn’t make any sense.”

For the EU27, leaving the details of the backstop out of the withdrawal treaty would not provide enough legal certainty.

“[The EU27] see no alternative,” an official from a major EU27 power said, adding that British concerns about “territorial integrity” are not justified.

To try to bridge the gaping hole between the two sides, the EU is prepared to include wording in the political declaration stating that the Irish protocol will never actually come into force, because the trading relationship both sides envisage signing will be so ambitious as to make it redundant.

To the U.K. this might be politically helpful, but it is legally “meaningless,” in the words of one senior official intimately involved in the negotiations.