Let us define two languages of Turing machines. $$ EQ_{TM} = \{<M_1,M_2> : L(M_1) = L(M_2)\} $$ $$ ALL_{TM} = \{<M> : L(M) = \Sigma^*\} $$ It is easy to show that neither of the languages are in $RE \cup coRE$, and also it is easy to construct a reduction $ALL_{TM} \leq EQ_{TM}$.
I suspect that there is no reduction in the other direction, but I don't know how to prove it. Is there some sort of hierarchy of undecidable/non-recognizable languages that proves it?