  Using a simple Gaussian-like Ansatz for the phase distribution of a theory
with a complex action, we show how the thimble integration for the average
phase factor can be plagued by a strong residual sign problem when the phase of
the complex integration measure conspires with the constant phase of the
integrand along the thimble. This strong sign problem prohibits the accurate
computation of the average phase factor when it becomes exponentially small,
and causes a strong sensitivity to the parameters describing the phase
distribution.
