PHX Tower Controller; providing OJT; experienced operational error at 1400 with two arrival aircraft in an overtake scenario; complicated by another developing event involving two departing aircraft.

Narrative

Providing training on Local Control South. Local Control South requested and received a release on a northbound turbo off Runway 26. When the turbo got airborne he was actually southbound. I took the position from my trainee and spent the next few minutes prying apart the turbo from a southbound air carrier as well as telling Local Control South what to do. Meanwhile a CL60 was inbound for Runway 25L with a PC12 in trail. The PC12 was faster and not visually separated from the CL60. As I was coordinating with Approach Control; the PC12 checked in on Local Control South frequency. I did not hear him; so I did not reply. Approach Control called Local Control South and said I can turn the PC12 to a 180 degree heading and 4;000 FT if I need it. I replied I wasn't even talking to him and confirmed this with my trainee. Approach never called back; so I assumed they applied the appropriate separation. On a 2 mile final; my trainee reached out to the PC12 and cleared him to land. The Supervisor pulled the tapes and called the operational error on Local Control South.