We show that jet emission from a Bose condensate with periodically driven interactions, also known as "Bose fireworks", contains essential information on the condensate wave function, which is difficult to obtain using standard detection methods. We illustrate the underlying physics with two examples. When condensates acquire phase patterns from external potentials or from vortices, the jets display novel substructure, such as oscillations or spirals, in their correlations. Through a comparison of theory, numerical simulations, and experiments, we show how one can quantitatively extract the phase and the helicity of a condensate from the emission pattern. Our work, demonstrating the strong link between jet emission and the underlying quantum system, bears on the recent emphasis on jet substructure in particle physics.