Activate Phyrexian Altar by sacrificing the persist creature, adding one mana of any color.
When the persist creature dies, its persist ability triggers, returning it from your graveyard to the battlefield with a -1/-1 counter on it.
When the persist creature enters the battlefield, its fabricate ability triggers, putting a +1/+1 counter on it.
Repeat.
By leveraging Cayth, Famed Mechanist to grant fabricate 1 to your persist creature, every entry onto the battlefield conveniently patches over its own mechanical flaw. When the persist creature enters with a -1/-1 counter, its newly acquired fabricate ability triggers to place a +1/+1 counter on it; thanks to Magic's counter-annihilation rules, both counters cancel out and leave the creature completely clean for the next loop. You then feed that exact same creature directly into Phyrexian Altar to generate your colored mana and trigger the cycle all over again upon its return.