Let's start from the block diagram of the phase servo loop.

iGp12 averages ADC input for 500 ms (number of samples is dependent on the RF frequency). Input offset (due to the non-zero mixer offset) is subtracted from that average producing the error signal. The error signal is multiplied by the loop gain and loop sign (+/-1). Resulting value is integrated, producing the delta signal. Delta is added to the longitudinal phase shifter setpoint, thus adjusting the local oscillator phase. The goal is to keep the beam signal and the local oscillator in quadrature, resulting in a phase detector output. Signal flow in blue on the block diagram is RF and analog, while the EPICS phase servo loop running at 2 Hz is shown in red.

It's important to understand that the phase servo loop, as currently implemented, only works in the multibunch regime, where a significant number of RF buckets are filled. With a single bunch, the change in the mean value is small and can be swamped by parasitic effects, resulting in loop runaway. The same can happen if the offset is not set right or if the total beam current is very small (ADC input signal is $I_b \sin(\phi)$, so if $I_b$ is small, there might not be enough range to adjust the mean).