Is it possible to photograph the milky way (or just any stars or stellar objects) when a full moon is out?
With regard to reasonably bright stellar objects: technically, yes.
In addition to the phase of the Moon, which determines the overall amount of light falling on the atmosphere above a specific location on the Earth's surface, photographing astronomical objects when the Moon is in the sky depends on several other factors. How much moisture is in the air in your location, both in terms of vapor and clouds? How much particulate matter (dust, pollen, pollution, etc)? How calm or turbulent is the atmosphere over your location? At what altitude are you located? Just as ground based light pollution is reflected and dispersed based on atmospheric conditions, so the light from the moon is spread across the sky. If you were in an extremely arid location, especially at high altitude so that the atmosphere is significantly thinner, you will be able to see and photograph dimmer objects than if you are at sea level under a very humid and turbulent sky. This is one reason why the world's largest and most expensive terrestrial telescopes are located in such places. In addition to lessening the effects of light pollution, those locations reduce the effect of the light cast by the Moon on the Earth. But they only reduce it, even when filters are used that allow specific wavelengths to pass while filtering most of the Moon's broad spectrum light. The most demanding deep sky work takes place at those locations on nights after the Moon has set or before it has risen.
Think about it. In Earth orbit the Hubble telescope can take incredible images of dim, far away objects without having to be in the Earth's shadow from the Sun and Moon. Even dim, far way objects that have a fairly small angular distance separating them from the Sun. The reason this is possible is because there is no atmosphere to reflect and spread out the light from the Sun in the space between the telescope and the dim object being viewed.
How brightly you expose also affects what is and what is not visible in a photo. The following photos were taken on a cold January evening when the atmosphere was fairly dry and the Moon and Jupiter were very close to one another. While not full, the Moon was about 2/3 illuminated.
This one is exposed properly for the Moon. Note dim Jupiter at the upper left.
A crop of Jupiter from another exposure that overexposed some of the highlights of the Moon. Even though several of Jupiter's moons are in the field of view, they are not bright enough to be visible when exposure allows details of Jupiter's atmospheric bands to be visible.
Another look at roughly the same field of view, but with exposure increased to allow the brightest of Jupiter's moons to be visible. Notice that surface detail of Jupiter is now completely blown out.
And a wider view of the same scene, exposed for the brighter stars and moons of Jupiter in the frame. Jupiter is in the center of the frame which is rotated ≈90° clockwise from the previous two closeups. Notice the lens flare from the Moon that was shielded by a makeshift hood extension when zoomed in on Jupiter. Also note that if the first photo above had been exposed at this level, the flare would have been visible in that photo as well. There were 12 stops difference in exposure between the first photo and this one. (Both were taken with a 400mm focal length. The first was cropped tighter).
If you were to add the much more humid air typical of a Summer evening in the location these were taken and a full Moon, nothing other than the Moon and Jupiter would probably be visible as the overall brightness of the sky from the diffracted moonlight would drown out the light from the stars, even if the lens was pointed to an area of the sky that didn't allow the Moon to cause lens flare. And none of the stars visible in this photo are as dim as most of the stars visible in impressive photos of the Milky Way.
The challenge of capturing the milky way with the full moon is that for the most part, the color of the starlight is a broad-spectrum object. Contrast with the background skyglow is critical for recording the milky way. Usually, people accomplish this by getting away from city lights and getting above the low-altitude murky air. Also, specialized contrast enhancing filters that block man-made and natural skyglow wavelengths are used.
This means that using nebula or light pollution filters will produce little contrast enhancement except to remove skyglow (which helps to show dark portions of the galaxy against star fields). The moon is also a broad-spectrum object, so it will ruin much of the contrast that the filters may provide.
Some people use narrowband filters to selectively control wavelengths that are recorded. The Wratten #29 filter is a classic that allows the color of Hydrogen Alpha (near 680 nm) to be passed along with other near-red light.
Using a filter like the one mentioned would produce better contrast for emission nebula (luckily the milky way has a lot of these) - even with moon glow - however, the starfields of the milky way would also be reduced in contrast, too.
There are other filters that have a tighter bandpass and would do a better job of rejecting the moonlight. These are usually specialized for use in a telescope and may not allow wide-angle lenses to be used effectively. Many astrophotos can be taken even in full moon conditions if tight filters are used with proper baffles and hoods to control off-axis light.
It depends on the amount of particles in atmosphere, pollution or dust or water/humidity. Particles in air reflect the light from moon practically blanketing the whole sky with thin haze.
In very clear/clean air the moon would not be an issue as long as you keep the moonlight off the front element of your lens. This you can achieve with a good lenshood, or placing your camera behind an obstacle, so that your camera is under shadow from moonlight.
But this is theoretical, since there is not many places with air clear enough. Perhaps in winter when temps stay below freezing point for long enough that all humidity has snowed down. Places like inland Alaska, I think. I have never tried this myself.