Songstress' Requiem (ALT+Tab)

Vladmir's Offering gives the lifesteal you need for survival. Plus, the Ring of Basilius will grant you 50% mana regeneration. (Belive me, you need it in early games). Like you said, if possible, let others like Anti-Mage to take Vladmir's Offering is battle.

In normal matches, especially tournament-like matches, Images are easily blown off cover, so the illusions only purpose now is to block and burn mana. Same goes for Crystalys and Daedalus on illusions, their purpose is to due damage, not confuse them.

Your build with the Poor Man's Shield and Arcane Boots is fine, especially if you're lane with someone who doesn't have Arcane Boots. The problem is that you don't have the mojo to survive long enough to get Heart of Tarrasque before you die.

After some discussions here and there, your build is fine and may be more possible with Vanguard because the effectiveness of Poor Man's Shield wanes during middle game. I have the same thinking when I first read about Manta Style for Naga Siren. After some battles, I realize that Manta Style offers survivability (due to AGI and ms), which is much the reason why people uses it.

To sum things up, the reason why I set Manta Style's priority higher than Heart of Terrasque is because manta offers more survival which sums up to an easier Heart of Terrasque.

I hope I have enlighten you XD. Thanks for the feedback, anyway!

Just asking, what's the point in getting Vladmir's Offering? It doesn't give many benefits to your illusion compared to the price (if you don't have at least 2 or more other melee Heroes in your team), it also reveals the real Naga because of the lifesteal.
Same for the Crystalys or Daedalus, as the crit number of the illusions will be much lower than the number of the real Naga and the enemies will spot the real one. I'd rather take a Monkey King Bar instead of it.

I usually go for a fast Poor Man's Shield into Arcane Boots/ Power Treads into Vitality Booster and Diffusal Blade + Heart of Tarrasque as soon as possible, because Vlad's doesn't give lifesteal to your illusions and the aura is pretty useless. Manta Style is a good choice, but unless you have to buy it to disspell some debuffs/DoT spells ( Mana Leak, Viper Strike, Shadow Strike...) the Heart has higher priority in my opinion.


EDIT - I just realized you listed your build as a "semi-carry/push" build: my suggestions were for the hard carry-oriented Naga Siren, but you might still find them useful.