Tranquility Rod

Tranquility Rod comes from the Lost Jungle and is tagged as a Quest Reward, so you earn it rather than buy it. The wiki describes a serene flute-shaped rod that makes musical notes with every catch and gives a 15% chance to bestow the Spirit mutation.

Its numbers are mid-tier. 100 Luck ranks 104th of 184 obtainable rods, Lure Speed of 65 ranks 107th, Control of 0.15 ranks 54th of 185 and Resilience of 30 ranks 51st. The weight limit of 8,000 kg ranks 70th of 117. Only Control is above the midpoint, which makes the minigame comfortable but does not boost rarity odds.

So the reason to use it is the Spirit mutation chance, not the stats. A 15% chance per catch is not small over a long session, and Spirit fish can sell better than the same fish without it.

The only other rod listed for the Lost Jungle is Azure Of Lagoon, so it is a small shop to compare against. Pick Tranquility Rod if you finish its quest anyway and want a no-cost rod with a mutation hook. For pure Luck, other quest-free rods do better.