Suppose I have a random variable $\theta=(\theta_1,\dotsc,\theta_n)$; where the $\theta_i$ might have pairwise correlations. I decompose it into $\theta=\hat\theta(\phi_1,\dotsc,\phi_k)$, where $\hat\theta$ is a deterministic function, and the $\phi_i$ are random variables such that each $\phi_i$ is independent from all others.
Does this operation have a name? "Independent decomposition" or something?
I think this particular property is unlikely to have a generally accepted name.
However, a more general concept is commonly called decoupling, when the joint distribution of a function of dependent random variables (r.v.'s) is represented as a mixture of distributions of this function of independent r.v.'s. See e.g. Wikipedia and the reference there to the book Decoupling: From Dependence to Independence by de la Peña and Giné.
You can always write down the joint distribution compositionally. In terms of a density function: $$f(\theta_1, \dots, \theta_n) = f_1(\theta_1)f_2(\theta_2 \mid \theta_1)f_3(\theta_3 \mid \theta_1, \theta_2)\dots f_n(\theta_n \mid \theta_1, \dots, \theta_{n-1}).$$ These conditional densities can be sampled from using inverse CDF transformations from independent random variates $\phi_1, \dots, \phi_n$. That is, $\phi_1 = F_1(\theta_1)$, $\phi_2 = F_2(\theta_2 \mid \theta_1) = F_2(\theta_2 \mid F_1^{-1}(\phi_1))$, etc.
This, I believe, would satisfy what you are looking for because the $f_1, \dots, f_n$ constitute a deterministic function of the $\phi$s. You give me a draw of independent $\phi$s and I can convert these sequentially into the $\theta$s.
This process is highly non-unique, because we can permute the order of conditioning.
You may also want to look up copulas.
