By Crispian Balmer

ROME (Reuters) - Italy's bank shares plunged on Tuesday, shaking the financial foundations of the euro zone's third-largest economy and threatening contagion to other EU nations.

The crisis could push Italy back into recession and, in a doomsday scenario, generate a Greek-type meltdown that Europe would find almost impossible to contain

Italy's banks are suffocating under a pile of non-performing loans and, adding to the growing sense of instability, Prime Minister Matteo Renzi has promised to resign if he loses a referendum in October on constitutional reform.

Recent opinion polls say he will fall well short.

"Italy faces a severe crisis that is exponential. This is not gradual and not linear," said Francesco Galietti, head of the Policy Sonar risk consultancy and a former finance ministry official. "The immediate trigger is the banking crisis."

Italy's bank sector index <.FTIT8300> has fallen 30 percent since Britain voted on June 23 to quit the European Union, bringing its losses so far this year to 57 percent. The euro zone banking stocks index <.SX7E> has dropped 22 percent and 37 percent respectively. On Tuesday, the Italian index lost a further 1.44 percent to trade around three-year lows.

Italy is politically and financially fragile, often described as "too big to save" in a crisis, so even though there is very little direct economic linkage between its banks and the Brexit vote, any global shock creates major tremors here.

"Italy is essentially the fault line of Europe," said a former IMF official, speaking on condition of anonymity.

"Both the public debt and the banking sector are on a powder keg, being maintained by a process of non-recognition of accumulated losses in the system that they keep rolling over. The real problem is that somebody has to take the losses eventually."

HUGE LOSSES

Immediate concerns center on Italy's third-largest lender, Banca Monte dei Paschi di Siena <BMPS.MI>, which has the highest ratio of bad debts to outstanding loans among listed Italian banks. It has been told by the European Central Bank (ECB) to slash these debts by 40 percent over three years.

Late on Tuesday market watchdog Consob said short selling of Monte dei Paschi shares would be banned in Wednesday's trading session. Such bans aim to prevent speculators from forcing stocks down by selling borrowed shares heavily and then buying them back at a lower price, pocketing the difference.

Rome is in talks with Brussels to devise a plan to recapitalize its lenders, including BMPS, hoping to use public money to stave off potentially huge losses for bank bond and shareholders - many of them ordinary retail investors.

Such a deal might require the bending of anti-bailout rules that the European Union adopted in 2014 to force investors and some depositors to share the burden of bank failures. Germany says the rules must be respected, but Italy says flexibility is needed to prevent possible bank contagion stemming from Brexit.

"A solution should be found quickly or the world's oldest bank (BMPS) could fail and bring down the rest of Europe's embattled banking sector with it. The EU needs to show flexibility or Italy could go under," said Andrew Edwards, CEO of British based financial company ETX Capital.

Highlighting these concerns, BMPS dropped 20 percent on Tuesday, bringing its losses this year to almost 80 percent.

One EU official told Reuters that on Monday night at a meeting of the Eurogroup Working Group, the body that prepares the meetings of eurozone finance ministers, national envoys had expressed "concern" for the situation in Italy.

However, a second EU official said there was little appetite to change or soften the bail-in rules, with opponents arguing that Italy had signed up to the rules so should abide by them.

SUICIDE

Italy did just that last November when it saved four failing lenders, but in the process wiped out some 430 million euros of junior debt held by 12,500 small savers. Many of these savers said they had no idea their investment was so risky and one pensioner committed suicide after losing his life savings.

Some 5 billion euros of junior debt is outstanding at BMPS and Renzi fears political and public turmoil if thousands more people are ruined because Rome has to follow stringent EU rules.

Opposition parties have raised the specter of a new sovereign debt crisis and Italian government borrowing costs edged higher on Tuesday because of the banking travails.

"The crisis in the Italian banking sector could have similar political consequences to those following the explosion of the spread on Italian sovereign debt in 2011," said Renato Brunetta, parliamentary party leader of Forza Italia (Go Italy!) party.

On that occasion, the then prime minister Silvio Berlusconi was eventually forced to resign.

Apparently undeterred by the poor opinion polls, Renzi has promised to go if he loses the October referendum on his constitutional overhaul that he says will end decades of revolving-door government in Italy. His departure would be taken badly by markets that have backed his reform agenda.

"Political instability would indeed cause financial instability," said Guntram Wolff, director of Bruegel, an influential Brussels-based think tank.

Italy cannot afford further financial upheaval.

It has the heaviest public debt burden in Europe after Greece - some 133 percent of gross domestic product - and is ill placed to pump funds into its crumbling banks, which are burdened by some 360 billion euros of doubtful and bad loans, equivalent to almost a quarter of the nation's GDP.

But it is also at risk of tumbling into a vicious economic circle. Without help, the ailing banks have little room to offer credit, thereby crimping Italy's sickly economy, which has barely grown since the birth of the euro in 1999.

(Additional reporting by Danilo Masoni, Paul Taylor, Silvia Aloisi and Francesco Guarascio; editing by Janet McBride and David Stamp)