= Solution
A <Dirac-to-Weyl gamma-matrix similarity transform> with the required signs is
$$
\boxed{U=\frac1{\sqrt2}\begin{pmatrix}I_2&-I_2\\I_2&I_2\end{pmatrix},\qquad\gamma^\mu=U\widetilde\gamma^\mu U^{-1}.}
$$
It is a <unitary matrix>, with $U^{-1}=U^\dagger=2^{-1/2}\begin{pmatrix}I_2&I_2\\-I_2&I_2\end{pmatrix}$. Multiplication gives
$$
U\widetilde\gamma^0U^{-1}=\begin{pmatrix}0&I_2\\I_2&0\end{pmatrix}.
$$
The matrix $U$ commutes with every $\widetilde\gamma^i$: their common block structure uses the same antisymmetric $2\times2$ block matrix, and the <scalar> blocks of $U$ commute with each <Pauli matrix>. Hence $U\widetilde\gamma^iU^{-1}=\widetilde\gamma^i$, exactly the spatial matrices specified in the <Weyl representation of the gamma matrices>. A <similarity transformation> preserves every <Clifford algebra> relation, so no new consistency check is needed for the transformed representation.
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