Yes, this was a real design, although I don't believe the designation was F-12F. The original F-11F Tiger was a good day fighter, but was not able to achieve supersonic speed in level flight and was at the end of its development potential. "Sageburner" was supposed to fix these problems by mating the F-11 fuselage with the J-79 engine. Since the F-4 Phantom and the F-8 Crusader were already in production and could do everything the upgraded Tiger could and more, the Navy decided not to pursue the project.
Likewise, since no F-11s were exported, ther was no incentive to get production underway. That said, I wonder what might have been if some early F-11s had gone into service and were then upgraded with the J-79 (in a process similar to waht the Israelis did with the Mirage III) and other enhancements.