WASHINGTON—With two months to go until Britain’s referendum on whether to leave the European Union, the bloc is determined to get its house in order. Nothing must be allowed to happen between now and June 23 that might boost the campaign for a British exit, or “Brexit”—an outcome that the International Monetary Fund last week predicted would trigger a “severe regional and global shock.”

The first step in this cleanup operation was last month’s EU-Turkey deal, an arrangement that for now, at least, has mostly stopped the flow of new migrants into the EU, defusing a crisis that was proving a gift to the Brexit campaign. But the next item of unfinished business is Greece’s latest standoff with its international creditors over its bailout program, which if left unresolved threatens to reach boiling point when a bond redemption falls due in June—just as Britons go to vote.

Despite efforts to break the impasse on the sidelines of the IMF’s spring meetings in Washington last week, the Greek bailout negotiations continue to run into the same stumbling blocks that have paralyzed the country for much of the past two years. Germany and its Northern European allies say they won’t lend more money to Greece unless the IMF agrees to lend, too. But the IMF won’t lend unless Athens commits to tough reforms and Germany agrees to allow Greece substantial debt relief, which neither wants to do. No one has yet found a way through what one EU official calls this “triangle of mistrust.”

Greece’s European creditors and the IMF did at least agree on one thing in Washington: how to resolve their differences over the targets for the next phase of the Greek program. Under the deal struck last summer, Greece is supposed to deliver a budget surplus before interest costs of 3.5% of gross domestic product in 2018. The European Commission believes that to reach this target, Greece needs to deliver further austerity measures this year equivalent to 3% of GDP. The IMF, which takes a more pessimistic view of the damage inflicted by last year’s standoff, believes that it will require extra austerity measures equivalent to 4.5% of GDP to hit the target, which it argued was unrealistic. The IMF had instead been arguing for a lower budget surplus target of 1.5% of GDP, which would have required extra austerity measures of only 2.5% of GDP, according to its calculations.

The compromise is about as brutal as it could be for Greece. Both sides have agreed to stick by the original 3.5% target and to use the commission’s forecasts as the baseline. But Greece will be required to commit to—and possibly legislate—the extra austerity needed to make up any shortfall, should the IMF’s more gloomy assessment prove more accurate.

What happens next depends on Athens. It must now agree on the measures necessary to deliver these targets—measures IMF insists must be “credible.” If the 3.5% target isn’t met, Germany won’t agree to debt relief, and without a guarantee of debt relief, the IMF can’t join the bailout. But the IMF’s definition of credible will create major political problems for Greece’s left-wing government. It has made clear it won’t accept continued reliance on the “soak the rich” approach to the public finances pursued by successive Greek governments that it argues have become self-defeating.

Although Greece has made a remarkable fiscal adjustment since the crisis, this has been largely delivered by piling new taxes on the same narrow base while cutting back on discretionary expenditure to protect public-sector jobs and wages and welfare entitlements.

Remarkably, 55% of Greeks are exempt from income tax, compared with 2% of Portuguese and 5% of Irish, according to the IMF. Meanwhile the Greek pension system costs Athens the equivalent of 10% of GDP every year, far more than any other country in Europe.

The IMF argues this is unsustainable: Tax-collection rates have gone down as punitive taxes on higher earners have fueled tax evasion and capital flight. Meanwhile, the economic and social fabric of the country is being wrecked by excessive cuts to spending on vital public services including essential equipment for schools and hospitals and maintaining infrastructure.

The IMF has long insisted that the only way credibly to strengthen Greece’s public finances now is to broaden the tax base and cut pension spending. It is hard to see how it could now credibly back down.

The carrot for Athens is that if it signs up to these measures, the country should receive its long-promised debt relief, paving the way for the normalization of financial conditions, including eligibility for the European Central Bank government bond-buying program and access to cheaper ECB bank funding facilities.

But as things stand, the IMF and Berlin are far apart both on what is a long-term realistic budget target for Greece—which will determine the amount of debt relief that is necessary—and the timing of any debt reduction. Berlin is concerned that, if given the chance, Athens will quickly start borrowing and spending again.

Can the triangle of mistrust be broken, paving the way for a deal in May? Only if Athens and Berlin are willing to give ground. Both will need to sacrifice considerable political capital. The alternative is that Europe finds itself again debating Grexit, even as it faces the very real possibility of Brexit.