  We present a new scheme which numerically evaluates the real-time path
integral for $\phi^4$ real scalar field theory in a lattice version of the
closed-time formalism. First step of the scheme is to rewrite the path integral
in an explicitly convergent form by applying Cauchy's integral theorem to each
scalar field. In the step an integration path for the scalar field is deformed
on a complex plane such that the $\phi^4$ term becomes a damping factor in the
path integral. Secondly the integrations of the complexified scalar fields are
discretized by the Gauss-Hermite quadrature and then the path integral turns
out to be a multiple sum. Finally in order to efficiently evaluate the
summation we apply information compression technique using the singular value
decomposition to the discretized path integral, then a tensor network
representation for the path integral is obtained after integrating the
discretized fields. As a demonstration, by using the resulting tensor network
we numerically evaluate the time-correlator in 1+1 dimensional system. For
confirmation, we compare our result with the exact one at small spatial volume.
Furthermore, we show the correlator in relatively large volume using a
coarse-graining scheme and verify that the result is stable against changes of
a truncation order for the coarse-graining scheme.
