For electrical safety, the soldering iron tip is to be earth-grounded. Exception to this is if the power supply is double-insulated (like a two-prong laptop brick) but these still have Y-caps and leakage current that will make their output have a potential, enough to damage IC's. If in doubt, just measure it, ACV between DC(-) and earth-ground.
Because the T12 clone power supplies are never tested or certified to any safety standard, I would never expect the pri-sec insulation to be safe. Are they double-insulated? How many layers of tape? We don't know, so I earth-ground the secondary.