This is a question about models, not a question about the real world.
We are used to invoking shareholder unanimity theorems, whether they are justified or not. But say the U.S. government owns twenty percent of each major bank. Exactly what instructions do they give the management? ("Hey, guys, just get stuff going again!"?) Presumably the twenty percent shareholder wants something different, and more in line with the public interest, than the desires of the remaining eighty percent. Are we to assume that the twenty percent wins out? Can managers be sued for violating their fiduciary responsibilities? Does the twenty percent explicitly tell the managers to do something other than maximize profit? What if the eighty percent votes to override them?
What are control rights worth in these situations?
You might argue that the mere fact of recapitalizing the bank will cause the eighty percent to want what the government shareholder wants. That is not in general true, especially if the government is pulling its capital back out at some point.
You might argue that the government involvement is a kind of insurance and it makes the older equity claims more like debt (insurance on the downside but loss of some potential upside gains). The newly neutered "debtholders" still might not want the bank to be very active, as evidenced by the stagnant nature of some explicit current debt markets.
I hear this recurring voice: "Hey, you guys, just get stuff going again!" It’s an odd basis for corporate governance.
I am not sure what is the proper way to model this set-up.
Addendum: Greg Mankiw proposes non-voting shares.