Less than 48 hours after Theresa May threw in the towel as British prime minister, admitting: “I have not been able to deliver Brexit,” her Tory party cohorts have been queuing up to replace her. Among the Conservative ministers and MPs who have thrown their hats into the ring are former Brexit secretary Dominic Raab, former leader of the House of Commons Andrea Leadsom, health secretary Matt Hancock, foreign secretary Jeremy Hunt, backbencher Esther McVey and international development secretary Rory Stewart. Boris Johnson and current environment secretary Michael Gove are preparing to face off once again.

But whoever inherits Mrs May’s job will also inherit most of the problems she faced in delivering Brexit, some of them made worse by the mistakes she made and the decisions she took.

At the heart of these problems is the legacy of the 2016 referendum. The Leave campaign assiduously avoided specifying any particular form of Brexit. All that was promised was that it would not be economically damaging, and that it would be quickly and easily achieved.

The latter set expectations that could not be met; hence the repeated refrain from Leave voters now is: “Just get on with it”. With Mrs May having been in office for just under three years, and having twice postponed the planned date to leave the European Union, her successor will be under enormous pressure to meet the current deadline of October 31.

The combination of the time needed for the leadership election and the summer political break means the new leader will really only have a couple of months to deliver.

And, like Mrs May, any successor will need to be mindful that voters were promised there would be no costs to leaving – even more so if the person chosen is one of those who made that promise.

As for defining what Brexit actually means, this continues to be a matter of bitter dispute. From the outset, the main options were a soft Brexit, meaning, primarily, membership of the single market, or a hard Brexit, which would mean leaving the single market and seeking a free trade agreement with the EU. These are entirely different economic and political models, but both require as a precondition a withdrawal agreement between the UK and the EU.

After some initial ambiguity, Mrs May opted for hard Brexit, with the UK leaving not just the single market but also the customs union, the system which exempts the UK and the rest of the bloc from tariffs on goods. It is inconceivable that the Conservative party will now elect as leader anyone who advocates a softer Brexit than this.

But a hard Brexit has a consequence which would destroy Mrs May’s attempts to ratify the withdrawal agreement she struck with the EU. By definition, being outside the regulatory area of the single market and the tariff area of the customs union means the UK having a hard border with the EU. And this would compromise the Good Friday Agreement that ended the armed conflict in Northern Ireland.

So in order to ensure that, whatever happens, there is an open border between Ireland and Northern Ireland, the withdrawal agreement specifies a backstop arrangement to be used unless another solution is found. This backstop would mean the UK as a whole remaining in the EU customs union and Northern Ireland also remaining in many aspects of the single market. That way, there would be no need for border checks.

Opposition to the backstop from the most pro-Brexit MPs in her own party and Unionist MPs from Northern Ireland, whose votes she needed for a majority, was the main reason why Mrs May failed to achieve support for her deal on multiple occasions and was forced to resign.

The key political conundrum is that these pro-Brexit MPs opposed Mrs May not because she failed to give them the hard Brexit they wanted but because in agreeing the backstop, she was asking them to recognise the practical realities inherent in a hard Brexit and the compromises needed to implement it. This they will not do.

That all roads lead to crisis reflects the fact that public debate is now so toxic, any outcome will be regarded by some part of the electorate as a betrayal

Instead, they want the backstop to be removed from the withdrawal agreement altogether, in favour of as yet unspecified or untested “alternative arrangements”, or to have a time limit on it (which by definition would mean it was no longer a backstop). If the EU will not agree then they propose to leave with no deal at all.

It is almost certain that Mrs May’s successor will come to office promising to pursue this approach. It is also likely that the EU will not accept the backstop being abandoned or diluted. Indeed it is not clear that they could do so without violating World Trade Organisation rules, even if the Good Friday Agreement was not an issue. So if the new leader sticks to the Brexiteers’ plan, there will be a no-deal Brexit (although some think the British parliament would find a way of preventing this). If they do not stick to the plan, the Brexiteers will revolt again.

But if the UK leaves with no deal, the Irish border issue doesn’t disappear. So in the event of a no-deal Brexit, there will immediately need to be new negotiations - about the border, trade and many other things – for which the EU’s precondition will be what was agreed in the withdrawal agreement, including the backstop.

This will bring things back to accepting something like Mrs May’s deal but with the addition of a major economic crisis. The only other possibilities would be to abandon Brexit altogether or shift to a soft Brexit – each of which would cause a major political crisis. For that matter, a no-deal Brexit would itself cause a political crisis since it is not remotely what Leave voters were promised in 2016.

That all roads lead to crisis reflects the fact that public debate is now so toxic, any outcome will be regarded by some part of the electorate as a betrayal. Mrs May’s main failure was that she did not try to build a consensus, difficult as that would have been, on a realistic way of doing Brexit until it was too late. Her attempt to hold cross-party talks ended in failure earlier this month and precipitated her departure.

Political discourse is now dominated by the language of sabotage, betrayal and treachery. Mrs May did little to challenge that and sometimes encouraged it. Positions on all sides have polarised. So even if a new leader were minded to seek consensus, the possibility of achieving it is very small. But in any case, it also means that the new leader is inevitably going to hold a position that will polarise things further.

In the end, the only way forward might be another referendum or perhaps a general election. But all the signs are that these would yield very close results and fail to provide any real resolution, for Mrs May bequeaths to her successor an even more divided country than the one she inherited.

Yet for all her faults and mistakes, the core challenge of delivering Brexit is not one of leadership but of Brexit itself, as her replacement will shortly discover.

Chris Grey is a professor of organisation studies at Royal Holloway, University of London and the author of The Brexit Blog