One of the first times I played Golden Sun TLA I found a code that imported information from Golden Sun, that appeared to either be highly optimised, or potentially modded. The giveaway was that there were two Cleric Rings available, though the wiki only says it is obtainable once, and must be transferred to TLA via a code.
Can I get another one by playing a NG+ cycle?
Quoting from https://goldensun.fandom.com/wiki/Password :
The Gold password is obscenely long, weighing in at 260 characters. Although this will be a pain to write down and fill in correctly, it does pay off enormously as you can transfer some items that you can't get in TLA, a notable example being the Cleric's Ring.
I'm not sure the data carryover from GS to GS:TLA counts as a "new game +", but if it does in your mind then you can carryover practically everything, Cleric Ring included.
So I misunderstood the capabilities of the game, which might be due to the fact that the wiki merges the functionality of the series, making it difficult to discern what is and isn't possible.
Quite simply; Golden Sun 1 does not have a New Game+ functionality. This is only available in Golden Sun: TLA.
