An explicitly promiscuous or flirtatious relationship, filled with sexual tension.
"The US and Canada have a special relationship because Canada is always on top"
"Josh has a special relationship with Emily from Poli Sci. She doesn't know he's a commy yet"
The current state of the "Special Relationship" between Trump's America and Brexit Britain, both having gotten largely out of hand in the last two years.
US: I don't know which one of us is dumber now, you with Brexit or us with Trump.
UK: Actually, we're both pretty fucked, mate. The Special Relationship has really become the Special Needs Relationship now.