There are lots of problems with thinking about monetary policy as setting a nominal rate of interest, and trying to keep the actual real rate of interest equal to the natural real rate of interest. But multiple own rates of interest (the "Sraffa problem") isn't one of them.
Suppose Canada produces and consumes apples and bananas. Suppose the Bank of Canada wants to target 2% inflation. And it wants to do this by setting an appropriate nominal rate of interest. Unless the relative price of apples and bananas never changes, and is never expected to change, any given nominal rate of interest will mean a different real own rate of interest on apples than on bananas. But so what?
There will be one time-path of the nominal rate of interest that is compatible with* keeping the inflation rate on apples exactly at 2%. But the Bank of Canada does not know what it is. And if the Bank of Canada ever gets off that "warranted" time-path, which it will, because it doesn't know in real time what it is, that warranted time-path will shift. Because monetary policy has real effects on savings and investment, as well as nominal effects on actual and expected inflation.
And there will be a quite different time-path of the nominal interest rate that is compatible with keeping the inflation rate on bananas at exactly 2%. It is quite a different time-path because all sorts of real shocks and real trends will cause the relative prices of apples and bananas to change over time. And if the Bank of Canada ever gets off that warranted time-path, that warranted time-path will shift, and it won't necessarly shift in the same way that the warranted time-path for keeping apple inflation at 2% would shift. Because monetary policy will have real effects, including real effects on the relative supply, relative demand, and relative price, of apples and bananas.
And there will be a third warranted time path for nominal interest rates, different again from the first two, if the Bank of Canada decides to target 2% inflation on a weighted average of the prices of apples and bananas, according to their weights in the CPI. And it too will shift differently from the first two warranted time-paths if the Bank of Canada ever misses that warranted time-path.
Look. Trying to keep the actual interest rate equal to the warranted interest rate is a problem. Because the Bank of Canada does not know what the warranted rate is, so will miss it, and because it doesn't know how its own misses will shift that warranted time-path.
But once the Bank of Canada has decided what price index it wants to target, the fact that targeting a different price index would mean it would be trying to solve a different problem with a different solution isn't itself a problem. Because it's not trying to target a different price index.
Multiple own rates of interest would only be a problem if the Bank of Canada decided to change its target price index. If it decided to switch from a 2% CPI inflation target to a 2% GDP deflator target, for example. It would have to change to a different time path for nominal interest rates. It would have to change how it responded to the consequences of its own past mistakes. And the Lucas Critique reminds us that the old empirical correlations and rules of thumb it had learned from experience under the old monetary regime might change or not work as well under the new regime. In fact, those old empirical correlations and rules of thumb could only be simply translated for use in the new monetary regime if money were always and everywhere neutral and superneutral, in the short run as well as the long run. And it isn't. Because if it were, then central banks could not use interest rate policy in any case, because the price level would be indeterminate.
Natural rates are a theoretical construct. Natural rates exist in natural rate models. A natural rate model is a model in which, in some sense (normally a "long run" sense), monetary policy is neutral and superneutral, so that some real variables in that model are invariant to some parameters (like the mean levels and mean growth rates of nominal variables) of the monetary policy regime.
What long run neutrality and superneutrality would imply, and what the meaningfulness of the natural rate concept does require, is that if the Bank of Canada had always been targeting 3% CPI inflation instead of 2% CPI inflation, the warranted time-path for the nominal interest rate in that counterfactual world would be everywhere 1% above the warranted time-path under a 2% CPI target. It's a very simple translation from one to the other. If that isn't true, then money is not superneutral, and you can't meaningfully speak of natural real rates that are independent of long run inflation. But comparing a 3% CPI target with a 2% CPI target is very different from comparing a 2% apple price target with a 2% banana price target.
In a natural rate model, there may be a natural rate of unemployment, a natural rate of output, a natural relative price for apples and bananas, and a natural real rate of interest. And they will all be changing over time, in response to real shocks and trends. Or, rather, there will be a whole vector of natural rates of unemployment, for all the different types of workers. And a whole vector of natural rates of output, for all the different types of goods. And a whole vector of natural rates of relative prices, for all the different goods. And a whole vector of natural real rates of interest too, measured in terms of all the different goods. And they will all be changing over time, in response to real shocks and trends.
So what if natural rates are a vector? Or rather, a matrix? If your monetary policy misses one, you almost certainly miss them all. And if your monetary policy hits one, it almost certainly hits them all. ("Almost certainly", because the relationship between those variables and monetary policy may not be monotonic.) Pick one, and try to hit it, if you must, and if you can. [Update: That bit was maybe a bit misleading, because it looks like I'm saying it doesn't matter what target the central bank picks. See my reply to J.V. Dubois below.]
Better yet, don't try. Forget real variables, even as very short run targets for monetary policy. Target a nominal variable. Like NGDP. It's got $ in the units. And central banks ultimately only control the $ unit.
(*Notice I carefully said "compatible with keeping the inflation rate at 2%", rather than "causing the inflation rate to stay at 2%". That's because of the indeterminacy problem. An equilibrium path for the price level implies an equilibrium path for the nominal interest rate, but the reverse is not true.)