= Solution
\b[For every nontrivial dimension $d\ge2$, no such <pure state> exists. The literal claim needs this qualification: for $d=1$, the only measurement outcome is necessarily the same for both parties.] The contradiction below requires allowing all complex <orthonormal bases>, as quantum theory does.
If equal outcomes were certain in the <computational basis>, all amplitudes with different labels would vanish, so the normalized joint state would have the form
$$
|\Psi\rangle=\sum_{j=0}^{d-1}c_j|j\rangle|j\rangle.
$$
Choose distinct labels $j,k$ and replace their computational <vectors> in both local bases by $|+\rangle_{jk}=(|j\rangle+|k\rangle)/\sqrt2$ and $|-\rangle_{jk}=(|j\rangle-|k\rangle)/\sqrt2$, leaving all other <vectors> fixed. The joint amplitude for the \i[different] outcomes $+,-$ is $(c_j-c_k)/2$. Certainty of equal outcomes would force $c_j=c_k$.
Now use the complex pair $|+i\rangle_{jk}=(|j\rangle+i|k\rangle)/\sqrt2$ and $|-i\rangle_{jk}=(|j\rangle-i|k\rangle)/\sqrt2$ in the same way on both systems. The amplitude for different outcomes $+i,-i$ is $(c_j+c_k)/2$, because the conjugated bra factors on the $|kk\rangle$ term multiply to $(-i)(i)=1$. Thus equal outcomes would also force $c_j=-c_k$. Combining the two requirements gives $c_j=c_k=0$. Every label belongs to such a pair when $d\ge2$, forcing the zero <vector> and contradicting <normalization>. This proves <impossibility of universal identical-basis perfect correlations>.
The maximally <entangled state> $d^{-1/2}\sum_j|jj\rangle$ is not a counterexample: it correlates matching labels when one party uses a basis and the other its coordinatewise <complex conjugate> basis. Requiring the two complex bases to be identical is different. This distinction also follows from <conjugate-state steering from a maximally entangled pair>; restricting attention to real bases would miss the contradiction.
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