There are lots of red giants and white dwarf stars orbiting each other out in our galaxy, and probably just as many in every other galaxy as well. Any set of binary stars usually forms at the same time, but since the lifetime of a star is tied very closely to its mass, if the two stars aren’t the exact same mass, then they reach the ends of their lives at different times.
A red giant star hasn’t completely finished its lifetime as a star just yet - it’s no longer burning hydrogen in its core the way our star is, but it’s still going - just more loosely held together, and burning helium instead of hydrogen. A white dwarf, on the other hand, has already reached the end of its pathway. White dwarfs are formed after the star runs out of helium to burn, and the outer layers of the star are lost into the formation of a planetary nebula. Only a central nugget of the former star remains after it loses its grip on its outer layers, which remains as a white-hot ball of atoms, slowly cooling to the temperature of the void of space. This is also the fate of our star, several billion years from now.
So a binary star system with a red giant and a white dwarf is a system where one star has already lost the majority of its mass into a planetary nebula, and the other star, being of a different mass, has progressed through its evolution at a different speed, and hasn’t come to the end of its helium burning phase just yet. If the stars are far enough apart, they will simply continue to orbit each other, until the red giant star reaches the end of its life and either creates its own planetary nebula and white dwarf, or explodes as a supernova, leaving behind a black hole or neutron star.
If the stars are close enough together, some other fun things start to happen. White dwarfs can begin to steal away some of the red giant star’s gas, pulling it in towards itself and growing in mass. After enough time, and after enough material has been thieved from its companion, the white dwarf can explode in either a nova (which is a surface explosion, which doesn’t destroy the star) or in a supernova (which is a complete detonation of the white dwarf). Not all supernovae are triggered this way; some are triggered by two white dwarfs colliding, and some are triggered by very high mass stars reaching an explosive end to their life. But novae seem to be exclusively the land of large companions and white dwarfs, and given that the white dwarf has already aged its way to the end of its lifetime, that means many of the companions will be red giants.
There’s at least one very well known red giant/white dwarf system in the night sky of the northern hemisphere. Unfortunately, it is found in the constellation Cetus, which has no particularly bright stars, but you can point your face in its direction if you choose. Mira, the star at the top of the page, is a red giant star plowing through the thin gas between stars in our galaxy, producing a shock wave and a long tail. But hiding in these images is a very faint white dwarf, visible to us in X-rays, pulling material in towards itself. If you want to go hunting for white dwarfs lurking around red giants, try looking for stars which have exploded in novae.
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