POTENTIAL CONFLICT WITH LTSS.

Narrative

THE ACFT I WAS FLYING HAD RECEIVED NUMEROUS CLRNCS WHICH WERE NEARLY NON-UNDERSTANDABLE ON HF. EACH TIME; WE QUESTIONED THE CLRNCS AND RECEIVED THE CORRECT ONES. WHILE AT FL350 WE PASSED A COMPANY ACFT AT FL370. WE PASSED A RPTING POINT AND MADE A ROUTINE POS RPT. SHORTLY THEREAFTER WE WERE SELCAL'ED AND ASKED IF WE COULD CLB TO FL370. WE COULD; AND SHORTLY THEREAFTER; WE WERE CLRED TO FL370. AFTER CLBING TO FL370; IT BECAME OBVIOUS THAT THE ACFT BEHIND US WAS ON OUR SAME ROUTING. WE HAD SUSPECTED THAT HE WAS PROCEEDING ON A DIFFERENT ROUTING BECAUSE OF THE ALT CLRNC AND ALSO BECAUSE HE APPEARED ON OUR TCASII TO BE ANGLING OFF AT A DIFFERENT DIRECTION WHEN WE MADE THE TURN AT OUR RPTING POINT. THE ACFT BEHIND US RPTED TO THE CTR THAT THERE WAS AN ACFT VERY CLOSE IN FRONT OF HIM AT HIS SAME ALT; THE CTR THEN CLRED HIM TO MAINTAIN FL390. THE CTR THEN SELCAL'ED US ONCE AGAIN AND ASKED US WHAT OUR NEXT ESTIMATE WAS -- WE RESPONDED 45 PAST THE HR -- THEY HAD THOUGHT IT WAS 52. THE ACFT BEHIND US WAS ESTIMATING THAT SAME POINT AT 49 SO NO MATTER WHAT TIME YOU GO BY; ATC DID NOT HAVE THE 10 MIN SEPARATION REQUIRED TO ISSUE US A CLRNC TO FL370. WHILE MINS WISE; WE WERE CLOSE; OUR TCASII NEVER SHOWED CLOSER THAN 10 MI.