If taxation were the only form of backing, things would be a little more strange. The currency could work, but taxes would have to be defined differently. Somewhere in the tax rules, there would have to be some statement that pins the value of the USD.
It is a fascinating problem though, because the price level is really just an arbitrary number when you look at it from a global perspective: If all price tags were removed from everything, an alien observer would have no way of telling it.
This makes a lot of people very uncomfortable, and I suspect that is what underlies a lot of the ultimately romantic desire to tie currency to something "real".
Presumably you'd control the value of currency through the same channels as under the existing system: indirectly by influencing interest rates, and instead of doing it by the Fed buying and selling bonds, you'd do it by imposing a new variable tax on leverage created by the banking system. It would be painful to adjust to (the base interest rate would be a direct cost rather than an opportunity cost, making banks' margins thinner) but ultimately work in a similar manner to the existing system.