Attosecond streaking, as a measurement technique, was originally conceived as a means to characterize attosecond light pulses, which is a good approximation if the relevant transition matrix elements are approximately constant within the bandwidth of the light pulse. Our analysis of attosecond streaking measurements on systems with a complex response to the photoionizing pulse reveals the relation between the momentum-space wave function of the outgoing electron and the result of conventional retrieval algorithms. This finding enables the measurement of the quantum phase associated with bound-continuum transitions.