A Tower Supervisor reported similar fix names JJANN and JJAMM resulted in an airborne conflict between parallel runway departures when one aircraft turned toward the fix opposite direction from ATC's initial heading.

Narrative

I was training on Flight Data (FD)/Clearance Delivery (CD)/Ground Control (GC) and issued the clearances and taxi instructions to both aircraft. Based on the configuration of the Tower cab; my On-the-Job Training Instructor (OJTI) was seated closest to the computer at the OS/Controller in Charge (CIC) when the Local Controller asked for the on course heading to JJANN. The fix was not spelled out but instead pronounced similar to the name Jan. The on course heading given by my OJTI was 298 which would have been a right turn off Runway 26R for Aircraft X. Concurrently; Aircraft Y was departing Runway 26L with a first fix of WLTER (a left turn heading 240) allowing more than the required divergence for the two. Aircraft X was told by the Local Controller to turn right on course and Aircraft Y was told to turn left direct WLTER. Aircraft X initially made a right turn observed from the Tower cab which is when the Local Controller switched them to Departure but turned left; toward Aircraft Y after being frequency changed. When the Local Controller saw the conflict; after no longer having either on frequency; he called Departure Controller and asked why Aircraft X was turning left when he was given a right turn and the Departure Controller said on course was left. After that call; my OJTI went to confirm the heading and said JJAMM and spelled out J J A M M and I heard him say M instead of N and verbalized that it was JJANN; which was actually a 209 heading and subsequently a left turn. Other factors; the Local Controller who requested the on course heading was a different controller than the one who issued the takeoff clearances due to normal position rotation and relief briefings. The oncoming controller was briefed the on course heading of 298 during the relief briefing. JJAMM and JJANN sound similar when spoken and even when spelled out (not phonetically) M and N sound similar. Additionally; while the pilot was instructed to turn right on course; they only read back on course and not the turn. Despite significant differences in geographic location; JJANN and JJAMM are similarly named fixes. Additionally; if there is a way with the equipment to have the on course headings from the departure point to the first fix on the flight strip; that would be incredibly useful in the terminal environment.