Example of Non-Equal Random Variables that are Identically Distributed? What is a simple example of two random variables $X$ and $Y$ that are identically distributed s.t. $X \ne Y$?
Is the only way to achieve this by changing the sample space underneath $X$ and $Y$ (i.e., $\Omega_X \ne \Omega_Y$)?

A method to generate such examples where the underlying sample space is the same is to use transformations that leave the probability measure invariant and apply them to the random variable. 
As Chris Janjigian mentions, one instance is when you have a random variable $X$ with symmetric distribution and let $Y:=-X$. 
Another, more specific example: let $X_i$ be independent standard normal variables, then for any unit vector $u$ (i.e., $\|u\|=1$) we have that
$$Y := u\cdot(X_1,\dots,X_n) \sim N(0,1)$$
This is also called rotational invariance of the normal distribution (because the joint density of independent standard normal variables is rotationally invariant). 
Examples where the sample spaces are different should be easy to construct (e.g., take two events with probability $1/2$ on different sample spaces and consider their characteristic functions).
