The aircraft were operating on the same air route within controlled airspace under the jurisdiction of Jakarta control sector. Although Jakarta was providing control and separation services within the Jakarta Flight Information Region (FIR), Perth Flight Information Service International (FIS INT) was monitoring the common international high frequency (HF) aeromobile facilities being used by both aircraft and Jakarta. G-BPFV was southbound at FL370 estimating LAMOB at 1914 hours and had established HF Primary Guard with Perth FIS INT. VH-EAM was northbound, 10 NM right of track, at FL350 until passing waypoint LAMOB at 1858 when a flight planned climb to FL390 was requested from Jakarta. FIS INT 1 intercepted Jakarta's approval for VH-EAM to climb to FL390 and recognised the potential for a traffic confliction with G-BPFV. FIS INT 1 immediately advised Perth control Sector 1 of the situation and provided traffic information direct to VH-EAM. Almost simultaneously, Jakarta instructed VH-EAM to maintain FL350 but the aircraft had already climbed to FL365 prior to resuming FL350. The aircraft had established the required 2,000 feet vertical separation again by approximately 1904, two minutes prior to the point of closest approach which was approximately 60 NM north of LAMOB, within the Jakarta FIR. It was later established that G-BPFV had been authorised by Jakarta to track 10 to 15 NM west of track. Neither the amended track for G-BPFV nor the change of level for VH-EAM were co-ordinated with Perth Sector or FIS INT for relay. There was a breakdown of the required separation standards.