  Non-perturbative study of "real-time" field theories is difficult due to the
sign problem. We use Bold Schwinger-Dyson (SD) equations to study the real-time
$\phi^4$ theory in $d=4$ beyond the perturbative regime. Combining SD equations
in a particular way, we derive a non-linear integral equation for the two-point
function. Then we introduce a new method by which one can analytically perform
the momentum part of loop integrals in this equation. The price we must pay for
such simplification is to numerically solve a non-linear integral equation for
the spectral density function. Using Bold diagrammatic Monte Carlo method we
find non-perturbative spectral function of theory and compare it with the one
obtained from perturbation theory. At the end we utilize our Monte Carlo result
to find the full vertex function as the basis for the computation of real-time
scattering amplitudes.
