Specifically, I find it confusing that your rows do not sum to zero because their elements can be denominated in different currencies under your notation.

You can just multiply the entire RoW column by φ before adding row-wise.

The whole thing would be much clearer (if also somewhat more cumbersome) if you had each of the three sectors hold both dollar assets and liabilities and reals assets and liabilities, and explicitly settled the FX transactions implied by the foreign and domestic sectors' willingness to hold non-native currency.

Well, it's already the case that the domestic sector holds both Real and dollar assets and liabilities. I'm also assuming the foreign sector has negligible willingness to hold non-dollar liabilities. Those assumptions are easiy relaxed by adding a couple of lines to the matrix.

I also have not exhibited explicitly the balance sheet. Maybe I should do that.

However, qualitatively I believe your conclusion rests upon the key assumption that the private sector's FX reserves operate broadly similarly to the strategic FX reserve. This is false: The private sector FX reserve is largely inaccessible during a national solvency crisis, and the strategic FX reserve is largely inaccessible in the daily operations of a non-crisis economy.

The only simplifying assumption is that the private sector's only foreign asset is cash, but the private sector does hold a foreign asset.

However, the stability condition involves only the domestic dollar liabilities, not the assets. So I am not assuming the domestic dollar cash is accessible in a solvency crisis.

What I am saying is that, if the domestic sector has a demand for foreign assets (cash), "clever" domestic-denominated derivatives won't fool them. So you cannot control the exchnge rate with derivatives.

A society committed to the notion that government is always bad will have bad government. And it doesn't have to be that way. — Paul Krugman