Since the problem is related to the sub panel, I would suspect a problem with the neutral wire between the main and sub panels. The two adjacent breakers verify that the two loads (refrigerator and lights) are indeed on opposite sides of the source. Breakers that have their handles tied together are protecting a 220V feed, such as a central air conditioner, clothes dryer or water heater, where each of the two breakers is monitoring one 120 volt side.
Imagine a 12v center tapped filament transformer feeding two 6v lamps. If both lamps are of the same wattage, the center tap would not be needed since each lamp would ,since being of the same wattage, have the same resistance and see 1/2 of the 12v. Think Ohms law and voltage divider circuit.
If, on the other hand, one lamp were twice the wattage of the other, without the center tap of the transformer being connected to the junction of the two lamps, the lower wattage lamp, having higher resistance than the other, would become very bright, even to the point of burning out. Again, think Ohms law and voltage divider circuit.
By connecting the center tap of the transformer to the junction of the two lamps, both would be lit appropriately to their own wattage since each would be receiving 6v.
Imagine a 12v center tapped filament transformer feeding two 6v lamps. If both lamps are of the same wattage, the center tap would not be needed since each lamp would ,since being of the same wattage, have the same resistance and see 1/2 of the 12v. Think Ohms law and voltage divider circuit.
If, on the other hand, one lamp were twice the wattage of the other, without the center tap of the transformer being connected to the junction of the two lamps, the lower wattage lamp, having higher resistance than the other, would become very bright, even to the point of burning out. Again, think Ohms law and voltage divider circuit.
By connecting the center tap of the transformer to the junction of the two lamps, both would be lit appropriately to their own wattage since each would be receiving 6v.