A philosophy of writing advocated by Rudy Rucker. A piece of transrealist fiction would be populated by characters who are lifted straight from actual people, generally people who the author knows. The narrator would always be the author, even if thinly veiled. In general, the work would be built around situations familiar to the author.
That explains the "realist" part. What about the "trans"? I'll quote from Rucker himself:
The tools of fantasy and SF offer a means to thicken and intensify realistic fiction. By using fantastic devices it is actually possible to manipulate subtext. The familiar tools of SF –– time travel, antigravity, alternate worlds, telepathy, etc. –– are in fact symbolic of archetypical modes of perception. Time travel is memory, flight is enlightenment, alternate worlds symbolize the great variety of individual world-views, and telepathy stands for the ability to communicate fully.*
How does this play out? Take Rucker's White Light: the narrator, "Felix Rayman"/Rucker, is experiencing a crisis of faith. Not just religious faith (although that does play a part), but faith in his position in the world, his abilities, et cetera. The entire set-up of the story is lifted directly from Rucker's own life at the time.
Through astral traveling, "Felix" leaves his body and enters a realm of mathmatical abstraction. In this realm, he is able to resolve his various problems. Hence, in White Light, Rucker uses the fantastical elements of his story to shed a different light on the more realistic elements.
Rucker's other transrealist novels are The Secret of Life, Spacetime Donuts, The Sex Sphere, The Hacker and the Ants and Saucer Wisdom.
*Taken from "A Transrealist Manifesto," in Rucker's collection of non-fiction, Seek!.