Final answer:
The vestibular reticular nuclei enable the body to maintain an upright posture by influencing the reticular formation and initiating spinal reflexes for posture and balance, forming part of the extrapyramidal system.
Step-by-step explanation:
The set of reticular nuclei that enables the body to maintain upright posture is the vestibular reticular nuclei. This is because the neurons in the vestibular nuclei project their axons to important targets such as the reticular formation, which influences respiratory and cardiovascular functions, and the spinal cord to initiate spinal reflexes involved with posture and balance. Specifically, the vestibulospinal tract incorporates equilibrium sensations with motor commands to support posture and movement. The reticular formation and vestibular nuclei are crucial components of the extrapyramidal system, which regulates balance, posture, and muscle tone without direct involvement of the cerebral cortex.