Sources familiar with the situation tell TMZ ... Khloe has not spoken to Tristan for several days because she's still furious after finding out he repeatedly cheated on her, and embarrassed her in the process.

We're told while Khloe's recent post made it appear as if she and Tristan were solid -- unveiling the baby's name as "our little girl," True Thompson -- sources tell us it's just a facade and their relationship is hanging by a thread.

We're told Khloe is nowhere near forgiving Tristan ... and the same goes for the rest of the Klan, including Rob, who is raging at Tristan. For now ... Khloe's putting on a brave face for the sake of their daughter. Khloe's sisters are also livid.

Khloe and the baby are staying at Tristan's house in Cleveland for the time being, largely because all of the baby gear is in the nursery there. But, they haven't been alone since Khloe was released from the hospital. There is definitely a chill in the air.

Kris has been by Khloe's side since the birth, and Kim, Kourtney and Kendall flew to Cleveland on Monday to support their sister after she had an emotional breakdown. But things will soon change. We're told once Khloe's medically cleared to fly, she's getting the hell outta Dodge and bringing True back with her to L.A.

Aside from being the baby daddy, Tristan's status with Khloe is tenuous at best.