Is there an analogue to the product measure construction for conditional independence?

Question

First I will sketch out the situation for the usual product of measure spaces and then ask whether the appropriate generalization exists for conditional independence. Finally I explain why such an object may be of interest for probability(or measure) theorists.

Given two probability spaces $(X,\Sigma,\mu)$ and $(Y,\Sigma',\nu)$, one can construct a $\sigma$-algebra $\Sigma \times \Sigma'$ with the property that it is the smallest $\sigma$-algebra such that the projection maps

$$\pi_1: X \times Y \to X$$

and

$$\pi_2: X \times Y \to Y$$

are measurable. Moreover, this is the $\sigma$-algebra we define the product measure $\mu \otimes \nu$ on, which has two notable properties:

1. The projection maps are measure-preserving(for example for $p_1$ this means that for $U \in \Sigma$, we have $(\mu \otimes \nu)(p_1^{-1}(U)) = \mu(U)$)
2. We can compute the measure for $U \times V$ (for $U \in \Sigma$ and $V \in \Sigma'$) by $(\mu \otimes \nu)(U \times V) = \mu(U) \cdot \nu(V)$

The second property says that the measurable sets of the two parts of the product have events that are 'independent'. It could also be interpreted by saying that the projections are independent random variables. Of course, this second property also implies the first property, but the first property is also desirable by itself.

Essentially, I would like a construction in the same spirit to work in a more general situation. While checking some technical details about the category of probability spaces with measure-preserving maps, one inevitably needs to try to understand how fibers of measure-preserving maps behave. This original motivation is not central to the question, but in case you are curious, the article that inspired asking this is here.

In the more general situation, instead of being given two probability spaces, one is given two measure-preserving maps between probability spaces $$p: (X, \Sigma_1, \mu_1) \to (U, \Sigma, \mu)$$ and $$q: (Y, \Sigma_2, \mu_2) \to (U, \Sigma, \mu).$$

The special case of the product measure will be recovered by letting $U$ be the one element set. Now instead of the product $X \times Y$, we consider the 'pullback'

$$X \times_U Y := \{(x,y) \in X \times Y | p(x) = q(y)\}.$$

Again, we generate a $\sigma$-algebra $\mathcal{E}$ on $X \times_U Y$ such that the projections

$$\pi_1: X \times_U Y \to X$$

and

$$\pi_2: X \times_U Y \to Y$$

are measurable. Now my question is the following:

1. Is there a probability measure on $X \times_U Y$ such that $\pi_1$ and $\pi_2$ are measure-preserving?
2. Let $s := p \circ \pi_1 = q \circ \pi_2$, then $s^{-1}(\Sigma)$ is a sub-$\sigma$-algebra of the $\sigma$-algebra we generated. Can we even choose the measure such that the conditional independence property $$\mathbb{E}[\pi_1^{-1}(A)|s^{-1}\Sigma] \cdot \mathbb{E}[\pi_2^{-1}(B)|s^{-1}\Sigma]= \mathbb{E}[\pi_1^{-1}(A) \cap \pi_2^{-1}(B)|s^{-1}\Sigma]$$ holds for all $A \in \Sigma_1$ and $B \in \Sigma_2$? Here, this is a conditional independence property for the projections.

One can see that this reduces to the properties stated at the beginning of the question for the usual product of measures. Now, in the technical context this question arose, it would completely suffice to find a measure satisfying 1. But it feels like that constraint does not uniquely determine the measure, since it is nowhere explicitly specified what the measure does when intersecting preimages of the projections. So by demanding property 2., I think we would essentially already completely specify the measure, so now the only question probably is whether this actually extends to a measure.

This question also seems to be more interesting than just being a technical question. Usually, product measure provides the prototypical way of constructing independent random variables explicitly. If one wants to similarly take the measure-theoretic study of conditional independence seriously, one should try to do the same thing for conditional independence. It would be nice to be able to explicitly construct conditionally independent random variables using this construction.

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There seem to be significantly bigger technical hurdles for this construction than for the product measure case. First of all, the conditional independence formula, unlike the usual independence formula, does not specify exactly what the measure of $\pi_1^{-1}(A) \cap \pi_2^{-1}(B)$ should even be. We can integrate that formula on both sides, but that still doesn't really tell us how the measure should be defined.

The author of the article linked above has written a paper, where in example 6.2. he claims to have found the required construction in the case where disintegration of measures works. First, let $\nu_1^x$ and $\nu_2^x$ be disintegrations of $\mu$ along $p$ and $q$, respectively. Then we can give the measure on $X \times_U Y$ by

$$\nu(C) := \int (\nu_1^x \otimes \nu_2^x)(C) d \mu(x),$$

where $\nu_1^x \otimes \nu_2^x$ denotes the product measure.

But it is a bit unfortunate that this construction works for product measures so nicely, but for conditional independence it relies on the incredibly technical concept of disintegration of measures. So a satisfactory answer to this question would explain whether for constructing this measure one inherently needs disintegration of measures or whether there is a more general approach working for all $\sigma$-algebras.