Given following equations
$a=m^{e_1}\mod n_1\\$
$b=m^{e_2}\mod n_2$
if you know the values of $a$, $b$ and all variables except for $m$, is it always possible to efficiently verify that the value of $m$ used to calculate $b$ was the same as the one used to calculate $a$?
If $n_1=n_2=n$ you could compute $a^{e_2}\mod n=b^{e_1}\mod n=m^{e_1e_2}\mod n$. However, if $n_1\neq n_2$, this does not work.