Step 1: Recall the exit route through the greater sciatic foramen.
Several pelvic nerves and vessels leave the pelvis through the greater sciatic foramen, below the piriformis muscle, and then hook around the back of the ischial spine and the sacrospinous ligament before entering the perineum through the lesser sciatic foramen.
Step 2: List the structures that make this hook around the ischial spine.
The pudendal nerve, the internal pudendal vessels, and the nerve to obturator internus all follow this exact path, passing behind the ischial spine on their way into the pudendal (Alcock's) canal in the perineum.
Step 3: Trace the obturator nerve separately.
The obturator nerve comes from the lumbar plexus and runs along the lateral pelvic wall, then leaves the pelvis through the obturator canal at the upper part of the obturator foramen, at the front of the pelvis, nowhere near the ischial spine.
Step 4: Compare the two exit routes.
The first three structures share one exit route around the ischial spine, while the obturator nerve uses a completely different anterior exit, so it is the odd one out.
Step 5: Final answer.
The obturator nerve does not pass behind the ischial spine.