It's essentially the two-seat trainer version. The F-5, likewise was being upgraded to become the F-20 Tigershark, but the program fell through...
Believe it or not the F-5/T-38 started life as a potential jet aircraft to be flown off of escort carriers. Most likely the Commencement Bay class, don’t know about the other, smaller classes. Then in the late fifties the USN decided there was no further need for any of the leftover CVE’s. Northrop then presented the N-156 as a possible advanced jet trainer. The Air Force chose it to replace the T-33’s