Let stratified $\sf Z$ have all axioms of $\sf Z$-$\sf Reg.$ but with Separation restricted to use only stratified formulas, where those are defined after Quine's New Foundations.

Define: $y = \mathcal P^{-1}(x) \iff x= \mathcal P(y)$

Define: $x= \bigcap_{s \in S} s \iff x=\{y \mid S \neq \varnothing \land \forall s \in S (y \in s)\}$

Now, lets try coin what we mean by rank or stage of the inverted iterative hierarchy:

A pseudo-ordinal is an acyclic transitive set of transitive sets. Where by acyclic set it is meant a set having no element of it being an element of itself.

$X$ is a rank, if and only if, there is a pseudo-ordinal $\alpha$ and there is a bijective function $f$ whose domain is $\alpha$ that sends $0$ to $X$ and that sends each $\beta+1 \in \alpha$ to $ \mathcal P^{-1}(f(\beta))$, and sends each limit $\lambda \in \alpha$ to $ \bigcap_{\kappa < \lambda} f(\kappa)$.

We phrase that as: $X$ is a rank of degree $\alpha$.

Axiom I: $X \text { is a rank } \land Y \text{ is a rank} \to X \subseteq Y \lor Y \subseteq X$

Axiom II: $\forall x \exists Y: Y \text{ is a rank } \land x \in Y$

Axiom III: $\alpha \text { is a pseudo-ordinal } \to \exists X: X \text{ is a rank of degree } \alpha$

Axiom VI: $\forall A \exists x \forall y \, (y \in x \leftrightarrow y \in A \land y \text { is a rank } \}$

Now, can this theory be consistent? I suspect the last axiom might cause problems, but even if so, can the theory without it be consistent? I mean this theory goes against the norms of the known cumulative hierarchy, where no descending power sequence like that is possible. So, does it stand a chance of being consistent whether with or without the last axiom.