They spiritbond. Most of the people I know that did a Relic spent weeks on it, because of the cost. It's rough, but it's pretty much the only option if you don't have the money to fail a bit of the rank IV's.
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The difference between using Tier IV materia and melding 3 stats is so small it's not worth worrying about. If you're concerned with the cost about the Tier IV materia, just meld a 3rd stat and move on to Nexus. You'll get MUCH better benefit from moving to your Zeta faster with all that sweet, sweet weapon damage instead of derping around about a few secondary stats that do almost nothing.
I have Longinus Zeta and Sasuke's Blades Zeta with capped DET/CRIT. This is what I did;
Level and invest in the appropriate crafting classes (with leves it's incredibly fast), and the zodiacs become effectively free/profitable. Then create your own gear to spirit bond with to get the materia you want. Meld whatever useless materia you get onto spirit bonding gear to increase spirit bond rate. Profit off the not-so-crap materia you don't need to get more materials to create more spirit bonding gear. After all is said and done for one Zodiac, you'll come out ahead.
Crafting five full sets (left and right side) takes me about 1.5 to 2hrs while watching a movie, because i'm lazy and crafting is boring (you could craft faster). Melding the crap materia I have or purchased off the MB for cheap takes about 30m for 5 full left/right side sets. With Spirit Bond potion, full melds, and FC buff, it usually takes about 2 hours to fully Spirit Bond all five sets, which give me 55 chances for the materia I want, of which I usually yield at least 1-3 Det/Crit V's.
It takes work and time, but it's supposed to. My group can meet for Final Coil only twice a week, so that leaves me with a lot of extra time on my hands (outside of the GF and full time job). If it wasn't for the Zodiac Quest, I'd really have nothing else to do that is progressive-minded.
Some people feel like they should be able to get a good weapon for less effort because they don't have enough time. Well, they can. They're called Primal/Soldiery weapons.
Quoted for truth.
You don't need to spend the extra amount/time on the last T4's, and the quest is forgiving enough to allow you to meld whatever off stat you want in order to not touch them until later (all the way to the completed Zeta phase), while benefiting from a higher iLv weapon immediately.
Yes, you still need a total of 75 infusions, so infuse whatever you want to complete after you've gotten to a stopping point on the attributes you actually want (weapon-depending, accuracy is not necessarily a bad stat to substitute with, it can potentially free up acc in other parts of your build).
It DOES consume alex per meld, but if you do your daily expert, you'll always have an alex per day from that quest alone. At the moment, i'm sitting on 77 alex and 3 extra maps I don't know what to do with (maybe if DRK has a relic/zodiac...), so you won't be wanting for more alex. If you are, there are a multitude of ways to acquire more (allied seals, alex maps with soldiery, FATE farming with Nexus and up).
You can respec, and getting Alex is pretty easy these days. The first melds you do aren't the stats you're stuck with forever, so it's more efficient to meld three useful stats now than do three weeks of spiritbonding for a "perfect" Crit/Det build when you could be working on Nexus at the same time.