I had an idea for a planet which, millions of years ago, was accidentally contaminated with various animals, plants and microbes from Earth, and - via the planet's native aliens' bizarre reproductive biology - the two independent lineages hybridized and produced fertile descendants.
Is there any conceivable reproductive mechanism, however alien, which could allow these organisms to interbreed with Earthly ones? One of the first problems is that of DNA - however, it's not statistically impossible that it would be used by the aliens too, coincidentally. Or maybe you can think of a way to do it without DNA.
One very hazy idea I had was something loosely based on the reproduction of lysogenic viruses, so you can try that if you want, but I didn't expand on it much more.
EDIT: In response to a comment on L.Dutch's answer; horizontal gene transfer, though not technically interbreeding, is fair game, as is any other method which produces chimeras rather than true hybrids.