Finding a joint distribution given a marginal and conditional distribution

Question

My question is given a marginal distribution $p_X(x)$ and conditional distribution $p_{X|Y}(x|y)$, am I guaranteed to be able to find a joint distribution $p_{X,Y}(x,y)$.
In almost every form of this question I have seen asked or discussed, the questions starts given $p_X(x)$ and $p_{Y|X}(y|x)$. I understand in this case, one simply must multiply these two, i.e. $p_{X,Y}(x,y)=p_X(x)\cdot p_{Y|X}(y|x)$.
In my case, I am considering the following $$p_X(x)=\int_Y p_Y(y)\cdot p_{X|Y}(x|y)dy=\int_Y p_{X,Y}(x,y)dy$$ This in my eyes boils down to, am I guaranteed to be able to find a $p_Y(y)$ that is a valid marginal distribution to give me the above equalities.

Answer 1

This paper talks about compatibility of conditional distributions. In particular, there is a quote in Section 10 that is relevant. I am attaching a picture.

This other paper that is referenced in the above quote talks about uniqueness of $p_Y$ in the context that you are considering, but does not seem to address existence.