The European Union’s central crisis is one of legitimacy. The EU is primarily a system of legally binding rules agreed among governments and enforced by a supranational court. But what happens when a sizable number of citizens decide EU rules are harmful to their interests—or worse, prevent national governments from fulfilling their duty to protect their citizens?

This crisis of legitimacy lay at the heart of the Brexit referendum, particularly regarding EU rules that require the U.K. to allow any EU citizen to live and work in Britain. It also lies at the heart of two major standoffs due to come to a head this week concerning three countries still grappling with the effects of the eurozone debt crisis: What sanction should the EU impose on Spain and Portugal for flouting the eurozone’s fiscal rules; And should Italy be allowed to inject public money into its banking system without applying EU bail-in rules to bondholders?

In both cases, the legal position looks clear-cut. Spain and Portugal missed their fiscal targets by a mile. Spain was supposed to deliver a headline deficit of 4.2% of gross domestic product in 2015 but managed only 5.6%, while its structural budget balance—which excludes cyclical factors— improved by only 0.6% compared with a target of 2.7%. Portugal’s headline deficit in 2015 was 4.4% compared with a target of 2.5%, and the improvement in its structural balance was just 1.1% compared with a target of 2.5%.

The European Commission will decide this week whether to uphold the fiscal pact—which was strengthened in 2011 after the previous version proved toothless—by imposing a fine of up to 0.2% of GDP and suspending their EU funds.

Similarly, Italy had three years to clean up its banking system before new EU bail-in rules requiring creditors to take losses on their holdings before any public money can be injected came into force in January. Yet European Central Bank stress tests to be published on Friday are expected to show serious capital shortfalls at several banks, including Banca Monte dei Paschi de Sienna, the country’s third-largest lender. The market seems unlikely to provide this capital. EU rules do allow public money to be used for “precautionary recapitalizations” where banks have failed stress tests, but only if there is also “burden sharing” by equity investors and junior bondholders.

All three governments have appealed for leniency. The Spanish government told the Commission that a fine would be “incoherent and counterproductive,” given Brussels’s past praise for Madrid’s for reform efforts and the strong recent performance of the Spanish economy. Portuguese Finance Minister Mário Centeno similarly questioned the wisdom of fining a country still facing deep economic challenges despite its compliance with a tough, Brussels-designed bailout program. Italian officials privately warn that imposing losses on junior bank bonds, many of which are held by retail savers, could lead to bank runs while also provoking a political storm ahead of an October referendum on constitutional reforms, in which the government’s survival is at stake.

All three countries blame wider eurozone policy failures for exacerbating their difficulties. Spain and Portugal say negative inflation has blunted the effectiveness of spending cuts. Italy has vocally criticized Germany’s failure to take steps to cut its 10% current-account surplus and boost eurozone demand. Madrid, Lisbon and Rome also point to rising hostility toward the EU—particularly in the wake of the Brexit vote— as a reason not to treat them harshly. “Adopting sanctions…would have a highly negative impact on the level of support for the European project, which has been largely consensual since 1976,” Mr. Centeno says.

Many economists are sympathetic. “Regional policy makers and politicians need to be more sensitive to how the complex, rule-based system that governs the region at the moment makes it less lovable,” says David Mackie, chief European economist at J.P. Morgan. “We think it would be very helpful in the current environment for the fiscal authorities and banking regulators to use the flexibility in the current rules.”

This flexibility could include the commission using its discretion to set the fines for Spain and Portugal at zero and using financial-stability provisions in the state-aid rules to spare bondholders from bail-in rules. There is, in any event, a solid case for looser fiscal policy in the eurozone right now, says Gilles Moec, chief European economist at Bank of America Merrill Lynch.

But excessive flexibility also brings risks. Both the fiscal and banking rules were designed to prevent fiscal risks spilling over among member states and were a crucial part of the political bargain that paved the way for the ECB’s bond-buying program.

Yet many policy makers privately concede that the fiscal rules have already been stripped of credibility, not least when Commission president Jean-Claude Juncker justified allowing France another two years to bring its deficit under 3% of GDP because “France is France.” “If the banking rules are hollowed out too, I think even I’d want to quit the euro,” says one German official.

The optimal outcome now may be a classic euro fudge that pays lip service to rules that may no longer make sense but are too difficult to change. That may limit the most troubling immediate economic and political risks—but at the cost of deepening the EU’s underlying crisis of legitimacy.