I think this is roughly what is going on in their model (pdf). I'm not 100% sure I'm right. And I'm not even trying to be 100% right. I'm trying to get the intuition (reverse-engineer) for those bits of their results I find most interesting.
Suppose you are really bad at algebra. You can't solve the equation X = 0.5X. So you make a tentative first guess at the answer, say X=1, plug your guess into the right hand side, get X=0.5, which is your second guess, which you plug into the right hand side again, to get X=0.25, which is your third guess, and so on. Eventually your guesses converge to X=0. It works for any initial guess.
We can think of this as tatonnement (groping) towards the answer, just like the Walrasian auctioneer who solves the supply and demand equations in micro by raising prices if there's excess demand, and cutting prices if there's excess supply.
But this tatonnement won't work if the equation is X = 2X. Unless you are incredibly lucky, and guess exactly right first time, your guesses will diverge further and further away from the right answer.
Consider a very simple one-period representative agent macro model, where each agent's choice X is a linear function of his expectation Xe of the other agents' choices. X = bXe .
This model has a unique rational expectations equilibrium, Xe = X = 0. But if agents try to solve for the rational expectations equilibrium by tatonnement, they will fail if b > 1.
Now let's look at the Neo-Fisherian question.
Let P be actual inflation, let Pe be expected inflation, Y be the output gap, i the nominal interest rate set by the central bank, and r the natural rate of interest.
A simple Newish Keynesian model would be:
P = Pe + aY where a > 0 (Phillips Curve)
Y = -c(i - Pe - r) where c > 0 (IS Curve)
Solving for P we get:
P = (1+ac)Pe - ac(i-r)
If the central bank holds i fixed, this simplifies to:
P = bPe + stuff, where b > 1
You can see the problem. If people try to solve for the rational expectations equilibrium using the tatonnement process, it won't converge, because b > 1. The rational expectations equilibrium does exist, and is unique, but there is almost zero chance the agents in the model will solve for it by tatonnement (unless they are extremely lucky and guess it exactly right first time).
But if, by sheer fluke, they did guess lucky first time, the solution is:
P = Pe = i - r
Yep, it's the Neo-Fisherian result: if the central bank sets a higher nominal interest rate, the result is a one-for-one increase in actual and expected inflation. But it's not a very sensible result if we only see it by sheer fluke.
If we want to make b < 1, so the tatonnement does converge, the central bank needs to set the nominal interest rate as a function of actual (or expected) inflation. And it needs to ensure that the nominal interest rate increases more than one-for-one with actual (or expected) inflation. That's the Howitt/Taylor principle.
It is important to understand that this Schmidt/Woodford tatonnement, just like the Walrasian tatonnement (strictly, Walrasian tatonnement with Edgworthian recontracting, so that offers to buy and sell are not binding until the auctioneer has found the market-clearing solution) does not happen in real time. This model does not say that actual inflation will increasingly diverge over time from the rational expectations equilibrium if the central bank pegs the nominal interest rate. Instead, we have to interpret this model as saying that the rational expectations equilibrium is implausible if the central bank pegs the nominal interest rate, because agents won't be able to solve for it by tatonnement.
Now the little model I have sketched above doesn't have any lags or leads (which is why I can ignore the time subscript). Each period is independent of past and expected future periods. It's not exactly like the Schmidt/Woodford model. Which is why I called it a Newish Keynesian model.
A strictly New Keynesian model would have expected future income on the right hand side of the IS curve. And would have expected future inflation on the right hand side of the Phillips curve. (And it might also have lagged inflation on the right hand side of the Phillips curve too, if there is inflation inertia in the model.)
My little model, with no lags or leads, will explode instantly if the central bank sets a fixed nominal interest rate for just one "period" (unless people guess exactly right first time). But if you introduced lags or leads into the model, it is possible the model will still converge, provided the central bank holds the nominal interest rate fixed for only a finite period of time, and is expected to act sensibly (obey the Howitt/Taylor principle) in future. Schmidt/Woodford use real numbers in their simulations to show this will in fact happen (they want to know what happens at the ZLB, and whether forward guidance on nominal interest rates helps).
I think the intuition is that agents will be able to figure out by tatonnement that inflation will return to target when monetary policy returns to normal, and if the period is short enough (so "the future periods" matter more than "the present period"), that enables them to figure out by tatonnement what inflation will be in the period just before monetary policy returns to normal, and so on, all the way back to the present.
What worries me most about their model is that (I think) it contains no backward-looking inflation inertia. If there is inflation inertia in the model, then a central bank that fixed the nominal interest rate too high, even if only for a finite period of time, might see a cumulative decline in inflation from which it would be impossible to recover. (Here is my own "model", which does have inflation inertia.)
Most people are not very good at algebra, the real world is a lot more complicated than any macro model, and people are all different. It's already stretching it to assume that people can solve the model by tatonnement in their heads, infinitely quickly, provided b < 1. Most of us mortals have to watch what happens, and revise our expectations in the light of experience, and what the central bank tells us it's targeting. It's maybe more sensible to worry about tatonnement convergence than to assume people can somehow all coordinate their beliefs by magic, but I think worrying about real time convergence makes more sense still.