 Ambient constructible power sets when the reals are not well-orderable

If the ambient power set of $\omega$ is not well-orderable and $a\in L$, then the full ambient power set of $a$ belongs to $L$ exactly when $a$ is finite. Finite set operations give the forward construction for finite $a$. If $a$ is infinite, its constructible well-order gives an injection $\omega \to a$. The induced injection of power sets would make the ambient power set of $\omega$ well-orderable if $\mathcal{P}(a)$ belonged to $L$.