Assume we have an elliptic curve $E$ with a Tate (or Ate,...) pairing
$G_1 \times G_2 \mapsto G_T$
Now the task is to find $g_1, g_1' \in G_1$ and $g_2, g_2' \in G_2$
such that the discrete logarithm $x$ is equal, i.e.
$g_1' = g_1^x$ and $g_2' = g_2^x$
That is easy, but now comes the hard part:
This has to be done without knowing $x$. Moreover, the constructor of the points has to provide a proof that he does not know $x$.
With just $g_1' = g_1^x$ alone, that would also be easy by hashing into the group $G_1$.