  We show that complex Langevin simulation converges to a wrong result, by
relating it to the Lefschetz-thimble path integral, when the path-integral
weight has different phases among dominant complex saddle points. Equilibrium
solution of the complex Langevin equation forms local distributions around
complex saddle points. Its ensemble average approximately becomes a direct sum
of the average in each local distribution, where relative phases among them are
dropped. We propose that by taking these phases into account through
reweighting, we can solve the wrong convergence problem. However, this
prescription may lead to a recurrence of the sign problem in the complex
Langevin method for quantum many-body systems.
