  We present a novel strategy to strongly reduce the severity of the sign
problem, using line integrals along paths of changing imaginary action. Highly
oscillating regions along these paths cancel out, decreasing their
contributions. As a result, sampling with standard Monte-Carlo techniques
becomes possible in cases that otherwise require methods taking advantage of
complex analysis, such as Lefschetz-thimbles or Complex Langevin. We lay out
how to write down an ordinary differential equation for the line integrals. As
an example of its usage, we apply the results to a 1d quantum mechanical
anharmonic oscillator with a $x^4$ potential in real time, finite temperature.
