Medium Transport; Low Wing; 2 Turbojet Eng
Air Traffic Control Fully Certified
Air Traffic Control Time Certified In Pos 1 (yrs) 1
Situational Awareness; Distraction; Training / Qualification; Workload
ATC Issue All Types; Conflict NMAC
Human Factors; Procedure
I was training a CPC-IT [Certified Professional Controller-In Training] on the North position with all other low scopes combined. I had not worked with this CPC-IT prior to this day. [Aircraft X] was handed off to us by the high scope requesting to fly the 15 DME arc to runway 25R from JODAB. The high scope had sent him direct JODAB to join the arc and we descended [Aircraft X] to 3;000 feet.There were multiple practice approaches and flight following request[s] with numerous untagged targets in the airspace. One of these untagged targets was northbound at 2;500 for for several miles. This target started a slow climb to 3;000 feet and ended up near [Aircraft X] as [Aircraft X] joined the arc. The CPC-IT was calling traffic elsewhere and gave [Aircraft X] a traffic alert once the targets had passed. [Aircraft X] notified the facility of a NMAC several days later.  Being as this was my first time with this CPC-IT; I let them go farther than I should have since I was not familiar with their limits. I should have jumped in sooner instead of letting the CPC-IT try to get caught up. A South controller was requested around this time to take the South half of our airspace to free up the frequency congestion as well as lower the number of aircraft being worked. We were dealing with other traffic calls and vectoring aircraft to the finals and did not see the untagged aircraft start a climb into [Aircraft X 's] path.
DAB Instructing Controller reports of a NMAC that was reported to the facility a couple of days after it happened. Controller was training and was busy with traffic. Developmental did not call traffic until the two aircraft had passed.