What would it mean, in this context, for a measure of risk to be "self-consistent"?

At the first pass, it would mean that possibilities would add up correctly.

Here's one example: give every investment, stocks, bonds, cash, commodities, etc. a distribution of possible returns. Then, every holding company also has a distribution of possible returns, obtained by adding up and scaling the distributions of the stock respectively held. So long as the market changes at a rate substantially faster than the holdings change, this is a self-consistent model of stocks in that the total amount of expected stock value remains the same. One can then examine the distribution of possible returns of a given investment and assess one's own position of risk.

The problem with this is how limited it is. It does not, for example, account for hedge fund pricing well at all, since in hedge funds the holdings change nearly as fast as the market does. It doesn't account well for the wiles of any particular, fast trading investor. It does, however, lend itself to a fairly simple model for static futures, forwards, options, and swaps.

Why is this line of reasoning interesting? The credit systems we used to have couldn't handle the complexity of the derivatives that were being created. By one method or another, bad investments, especially mortgages, were being added up, and in the process the market found it difficult to track how risky the summed investments were. Credit rating agencies don't even make publicly known what their credit ratings represent! An investor is compelled to then blindly believe them, which, in this case, spectacularly failed.

How does this concept differ from the stochastic calculus based implementations of VaR?

I don't know. Maybe it doesn't differ from it at all. I'm not a domain expert by any means...

Thanks for the tip.

I've been thinking of applying agent based simulations as a new way to measure risk. That's a bit different than what you're thinking of but I think it would be self-consistent, or at least the way I understood it.

Agent based simulations would probably be a very excellent approach. Unfortunately, the system is highly nonlinear and quite chaotic, so you'd have to do some work into engineering dissipation into the system to keep it from fluctuating wildly.