Shinryu is an Original Creation monster bred at the Monster Arena in Final Fantasy X. The monster resembles the Guardian of Bevelle, Evrae, fought earlier in the game. This fight is held underwater, and as such, only Tidus, Wakka and Rikku can participate. Shinryu is unlocked after at least two of each of the underwater fiends in the Mt. Gagazet caves are captured.
Shinryu counters hostilities with a physical attack; and often uses Eraser, an attack which petrifies (unblockable even with a Stoneproof or Ribbon armor equipped) and shatters one victim instantly, removing them from the battle. Shinryu will stop using this attack if there is only one party member left. Another dangerous attack is Shining, which inflicts heavy non-elemental magic-based damage with 8 hits in random characters.
Every defeat of Shinryu will result in a weapon with Double AP or an armor with Auto-Med and two Wings to Discovery, needed for customizing Break HP Limit on armor. Its equipment drops provide a 6% critical hit chance bonus instead of the usual 3% (this is relevant to weapons and armors, but only to attacks that deal physical damage).
The best way to approach this is to ensure that stats have been maxed (with the exception for Luck and Magic, and HP of 9999 or over; a large HP count isn't necessary). This way, Shinryu's counter attack and magic attack Shining will do less than 1000 damage even without casting Shell and Protect.
The party should start using Hastega, and hack away using Quick Hits, which will be countered. If Rikku, Tidus and Wakka have obtained their Celestial Weapons and the party is dealing 99,999 each hit, the battle is not much trouble. The concerning factor of this battle is Shinryu's Eraser attack, which despite a character having Stoneproof or Ribbon customizations on their armor, will turn a character into stone and shatter them, removing them from the fight. However, after two characters have been erased Shinryu will not use Eraser on the last one, in which position it would be ideal to cast Auto-Life on the last character and keep using Quick Hit.
Shinryu is the Japanese name for Shénlóng, literally "god dragon" or "spirit dragon", an important ancient dragon in Chinese mythology. It is said to preside over storms and rain, and can bring disastrous drought, thunderstorms and general bad weather if it feels neglected.