Stochastic processes with same distribution, which are no modification

Question

Consider stochastic processes $\mathcal{X}$ and $\mathcal{Y}$. How can I find an example in which $\mathcal{X}$ and $\mathcal{Y}$ having the same distribution but are no modification of each other? I think I understood the concept of modification. But how do I find the distributions for $\mathcal{X}$ and $\mathcal{Y}$? I know that there can be defined a distribution by kolmogorov's extension theorem, but I wonder how this should look like. Any hint or help will be appreciated. Thanks in advance!