When you get to level 50 and complete all your job quests, you can start the "Relic Reborn" quest chain for your class.

The initial relic weapon doesn't glow, but upgrading it to Zenith gives it a glow, and just requires 3 Thavnairian Mists (150 soldiery each from Auriana in Revenant's Toll).

After that you're looking at an increasingly massive wall of grinding to upgrade your relic through various stages. The next stage (Atma) requires farming low droprate items from FATEs in a bunch of different zones, and that changes the base model of the weapon slightly but removes the glow. The next stages require increasing amounts of annoying grinding and busywork but make it glow more vibrantly as you upgrade it through each stage. The penultimate step upgrades it to a Zodiac weapon with a new model, and the final step upgrades it to a Zeta weapon with the most intense glow.