  We develop a stochastic resolution of identity approach to the real-time
second-order Green's function (real-time sRI-GF2) theory, extending our recent
work for imaginary-time Matsubara Green's function {\em J. Chem. Phys.} {\bf
151}, 044114 (2019)). The approach provides a framework to obtain the
quasi-particle spectra across a wide range of frequencies as well as predict
ionization potentials and electron affinities. To assess the accuracy of the
real-time sRI-GF2, we study a series of molecules and compare our results to
experiments and to a many-body perturbation approach based on the GW
approximation, where we find that the real-time sRI-GF2 is as accurate as
self-consistent GW. The stochastic formulation reduces the formal scaling to
$O(N_e^3)$, where $N_e$ is the number of electrons. This is illustrated for a
chain of hydrogen dimers, where we observe a slightly lower than cubic scaling
for systems containing up to $N_e \approx 1000$.
