Suppose a large Eurozone country – let’s call it France - decided that it needs to substantially increase its minimum wage in order to reduce poverty. The increase is sufficiently large that it leads to a sustained increase in average French wage inflation, which in turn decreases the competitiveness of France relative to the rest of the Eurozone. France cannot be permanently uncompetitive, so the obvious consequence would be that France has to endure a subsequent period in which its relative inflation was below the Eurozone average.
However this would require a period where French unemployment was above its natural rate. French politicians declare that this would be politically unacceptable to French voters. Instead they suggest French inflation should remain at 2%, and the remainder of the Eurozone should increase their inflation rate to 4% for a time (giving an average Eurozone inflation rate of over 3%) to ensure France regains competitiveness. Now this would not normally be possible, because the ECB’s inflation target is 2%. However the influence of France on the ECB is such that the ECB fails to raise interest rates in time to prevent 3% average inflation, and subsequently keeps interest rates low because they repeatedly forecast inflation falling back down to 2% in due course.
The rest of the Eurozone would understandably be upset at having to endure 4% inflation. Some countries might suggest that perhaps, in the absence of ECB action, they could tighten fiscal policy to get their inflation below 4%. However France refuses to countenance changes to agreed fiscal targets, and instead suggests that what is really required is for other countries to adopt a similar increase in the minimum wage to the one originally undertaken in France. The French head of the ECB gives a speech where he intimates that the ECB might be prepared to raise interest rates a little bit in exchange for other countries introducing this ‘structural reform’ to their minimum wage levels. The French government also hints that it might be prepared to allow very limited fiscal contraction outside of France, but only if this took the form of tax increases rather than public expenditure cuts.
Your reaction to this little imaginary story is that it couldn’t possibly happen because other Eurozone countries would not permit it to happen. My suggestion is that Germany rather than France is doing exactly this at the moment, except that in their case it started with a period where German wage inflation was below the Eurozone average (for reasons discussed by Dustmann et al here).  German control of the ECB might not be as complete and simple as I imagined French influence in the story above, but it has the advantage that interest rates have hit the zero lower bound, and the threat that anything unconventional could be declared illegal. And in this real world story I too wonder why other Eurozone countries allow Germany to get away with it.
 In fact what Germany is doing is worse, because inflation asymmetries and debt deflation mean that the output costs of achieving zero inflation outside Germany to regain non-German competitiveness are far greater than the costs associated with 4% inflation in my story.