The Shattered Man Glitch occurs in EarthBound when only one of the Shattered Man enemies in Summers' museum's heiroglyph room is defeated, leaving the second still alive. After defeating Giygas, the solitary Shattered Man is the only enemy that can still be fought. Losing against this enemy results in the game's graphics temporarily glitching and both Ness and Paula continuing despite the fact that they both lost, as there are supposed to be no enemies after beating the game and therefore no ways to be defeated. This glitch was removed in Mother 1 + 2.
The Starmen.Net walkthrough lists this glitch here (#11 on the list of Battle-Related Tricks), and a video of it can be seen here. While the two sources conflict on the results of performing the glitch, it appears that the walkthrough is incorrect in saying that the game freezes. When you press the Y button bringing up the map, you can exit it and appear as a ghost in the map. You can actually heal yourself in the hospital.