Obviously I've made an elementary mistake.
Consider a sine wave with amplitude 1 from a signal generator and assume that it takes 1 unit of power to make.
Consider it together with a second identical sine wave of the same phase. So total power to make is 1 + 1.
Add the two sine waves together. The result has amplitude 2. Power is proportional to amplitude squared so the total amount of power it carries is 4.
So how is it that I can put 1 + 1 units of power in and get 4 units of power out?
Or of the two had opposite phases how can 1 + 1 = 0?