The double tragedy has had a rippling effect ... complicating funeral plans. Family sources tell us when Carrie died the Coroner contacted her daughter, Billie, for permission to conduct an autopsy because she's the next of kin. We're told Billie -- who was distraught -- deferred authority to her dad, CAA honcho Bryan Lourd.

The family sources tell us their attorneys contacted the Coroner and expressed some concerns about moving forward, though they would not be specific. We do know families often contact the Coroner and ask that they not perform an autopsy, often for religious reasons.

We're told for now the Coroner is just waiting. In addition to the autopsy, the Coroner needs to know about what happened on the United jet when Carrie's heart just stopped beating, what meds she was on and her medical history ... in order to determine cause of death.

We're told when Debbie died Wednesday ... it threw the entire family into chaos and it has further complicated the manner in which Carrie's body will be handled.