This image below is from this TI cookbook.
Please look at the red box.
Question: why large signal can make a conditionally stable system unstable?
'large signal bounds' refers to non-linear effects, such as saturation (or 'limit'), where the loop gain is essentially reduced since the saturation element's gain effectively decreases as it's input signal amplitude increases; i.e. the input amplitude increases but the output amplitude remains at its limit, hence gain reduces.
When the circuit is operating at design gain, it's stable.
If any amplifier is saturated due to overload or as they are during power-up when their rail voltage is very low, then the gain is lower, and the loop becomes unstable. One danger is that instability during power-up results in overloads, which maintains the low gain and the instability. Another danger is is that an overload can happen, perhaps due to some transient being input to the system, that keeps the system oscillating between the end-stops.
A conditionally stable system needs some mechanism to detect overload, and to reset the operation back to the stable region.
The red box text refers to changes in load (not a "large signal" as per the OP) so my answer is an attempt to show how phase shift can change problematically when loads are added to the output of a current mode control loop voltage regulator. For this I'm using a gain of 1000 and a low pass output stage formed by a 10 k resistor and 1 uF output capacitor: -
As load changes the phase response is shifted higher in frequency (as can be seen) and, if the control loop relied on the phase shift being significantly close to a lag of 90 degrees for stability (at the 0 dB gain point) then as load is added this may cause more instability or less instability dependent on the control system make-up.