It might be worth to add a minor point to @Rory Alsop answer which I would like to be a comment (in light of other comments there) - but little too long for that.
In the answer we read quote:
At the beginning of Winter, when it's nighttime all of the time, the moon would be in the sky for the 2 weeks closest to Full Moon, and then below the horizon for the next 2 weeks.
That is basically correct, yet if one seeks accuracy, the "beginning of the winter" (for the Moon) should not be taken as exactly 21 March for the South and 22 September for the North pole. Rather this date moves across cycle of 18 years due to the Nodal precession of the Moon and the tilt of 5 deg from the ecliptic.
Let's consider the South-Pole. during the Summer when the RA of the Sun is between 180 (22 Sep) and 360 (21 Mar), we would naturally expect not to see a full-moon. This is the intuitive thought as the Moon should be 180 deg apart from the Sun (which during the Summer is always above the horizon), hence to be below the horizon.
But, due to the 5deg tilt we can see the Full Moon and the Sun both above the horizon! in order for the Moon to be full it needs to be 180 ecliptic longitude away from the Sun, which is also exactly 180 in RA (this 180 to 180 mapping between the coordinate system should not usually work. it works here specifically because it is 180 deg).
During in the 18 year cycle the Moon is expected to have the RA of its node move ~13 deg from each side of the [equatorial] node. If we translate this to the movement of the Sun it gives us ~13 days movement (to each direction) from the day we can see the Full Moon - so in this sense the "beginning of Winter" can move few days (~13) back and forth across the circle of 18 years.
Let's see concert example. Just before the March equinox, say in RA 355, when the Sun about the set in the South-pole, but still above the horizon; and if we are lucky enough to have a Full Moon that date, it means the Moon will be at RA 175. As we said earlier the Moon might have negative declination even from RA 167 (180-13). So at RA 175 the Moon might have negative declination and hence just little above the horizon in the South-Pole.
This will happen, for example, on 19 March 2030 at the South-pole. Both the Sun and the Full Moon are above the horizon. (though the Sun is quiet close to the horizon, the Moon is quiet far +4 deg of altitude (= (-4) in declination) - which demonstrate the point nicely).