So, let's suppose we have a causal diagram like this:
The $X\to Y$ arrow means $X$ definitely causes $Y,$ and the same for $U_1$ causing $X$ and $U_2$ causing $Y.$ The dotted line between $U_1$ and $U_2$ stands for an unknown causal relationship, but a correlation at the very least. Now, by assumption, $U_1$ and $U_2$ are unobserved. So, as it stands, we cannot be certain that the data we have and the calculations we perform are capturing the true causal relationship between $X$ and $Y.$
There are at least a couple possible solutions. One method is the front door adjustment, which goes like this: suppose you could identify and insert a mediator variable $Z$ in-between $X$ and $Y,$ like this:
Then the front-door adjustment formula is
$$
\newcommand{\doop}{\operatorname{do}}
P(y|\doop(x))=\sum_zP(z|x)\sum_{x'}P(y|x',z)\,P(x').
$$
This allows you to find the true causal effect of $X$ on $Y.$
Another approach is to use an instrumental variable, which we tack on to the back, like this:
What you would then do is:
We regress $X$ and $Y$ on $Z$ separately, yielding the regression equations $y=r_1z+\varepsilon$ and $x=r_2z+\varepsilon,$ respectively. Since $Z$ emits no backdoors, ... $r_1$ equals the total effect of $Z$ on $Y...$ Therefore, the ratio $r_1/r_2$ provides the desired coefficient... - from Causal Inference in Statistics, A Primer, by Pearl, Glymour, and Jewell.
