Let's start with a "normal" country, where there is one monetary authority that sets the money base M, and one fiscal authority that sets the fiscal deficit F. And aggregate demand (N for NGDP) depends on both M and F:
1. N = M + F
And suppose the monetary and fiscal authority agree on a quadratic Social Welfare loss Function, where the optimum N is N*, and the optimum F is F*:
2. Minimise L = (N-N*)2 + (F-F*)2
The equilibrium, trivially, is N=N*, F=F*, and so M=M*=N*-F*
But what happens if the monetary authority wants smaller NGDP than the fiscal authority? So N*m < N*f. (They might disagree on the Social Welfare Function, or they might disagree about shocks hitting the aggregate demand function).
If monetary and fiscal authorities move simultaneously, the Nash equilibrium (as shown on the diagram) is the solution to the fiscal authority's reaction function (3), and the monetary authority's reaction function (4):
3. F = (F* + N*f - M)/2 (Set F to minimise L taking M as given)
4. M = N*m - F (Set M to minimise L taking F as given)
And the equilibrium is:
5. F = F* + (N*f - N*m) and N = N*m and M = N*m - F* - (N*f-N*m)
This is a bad equilibrium. The monetary authority gets the level of NGDP it wants, but the policy mix is wrong. The fiscal deficit is too high, and the monetary base is too low.
If the monetary authority moves last, we get to a better (Stackelberg) equilibrium. The fiscal authority moves first, and picks a point on the monetary policy reaction function. The monetary authority takes F as given, so sets M to ensure that N=N*m. The fiscal authority recognises the monetary authority will do this, and so recognises it cannot affect N, and so sets F=F*.
Look at my diagram, to see the difference. If the monetary authority moves last, and takes F as given, it will set M to ensure that M+F=N*m. The fiscal authority knows it will face the vertical blue Aggregate Demand curve, where fiscal policy has no effect on NGDP. But if they move simultaneously, the fiscal authority takes M as given, and so faces the green upward-sloping Aggregate Demand curve, where fiscal policy affects NGDP.
[Notice, by the way, that my diagram is formally equivalent to the familiar Kydland-Prescott rules vs discretion diagram for inflation.]
In any normal country, with one fiscal authority, that fiscal authority will recognise that it will be on a higher indifference curve in the Stackelberg equilibrium than in the simultaneous Nash equilibrium. So it will let the central bank move last, to get to that Stackelberg equilibrium. It will ignore the effect of fiscal policy on NGDP, and let the central bank alone choose NGDP, even though it would prefer a higher level of NGDP than what it knows the central bank will choose.
But the Eurozone is not a normal country, because it has many fiscal authorities. The monetary authority can only set M and NGDP for the Eurozone as a whole. If each fiscal authority has its own region, it can use fiscal policy to increase NGDP for its own region, while recognising that the ECB will ensure that NGDP for the Eurozone as a whole is where the ECB wants it to be. Each individual Eurozone fiscal authority will face an upward-sloping Aggregate Demand curve, even though ex post the Aggregate Demand curve will be vertical for the Eurozone as a whole. (The exact slope of that Aggregate Demand curve will depend on the size of each individual country, and on how open its economy is.)
The normal country's Stackelberg equilibrium requires the monetary authority to be the Stackelberg follower and the fiscal authority to be the Stackelberg leader. The problem with the Eurozone is that you can't have 18 Stackelberg leaders. Each individual country, facing an upward-sloping Aggregate Demand curve at the Stackelberg equilibrium, has an incentive to defect from the leadership position, and become a follower.
The ECB cannot move last, because the 18 fiscal authorities have no individual incentive to move first. So the Eurozone is stuck at the undesirable simultaneous moves Nash equilibrium. That's why the Eurozone is a mess.
(This is a simple model, and like all simple models it leaves stuff out. That's the point of simple models. Please pay more attention to what it brings in, and less attention to what it leaves out.)
[P.S. Just a bit of Canadian content: To the best of my memory, the last time any Canadian province tried to move last, and use provincial fiscal policy to increase provincial aggregate demand, was Ontario under Bob Rae's NDP government, in the early 1990's. The attempt was recognised as a failure. Ontario is a very open economy, so it would face a fairly steep green AD curve.]