= Solution
At each qubit, two matrices from $\{I,X,Y,Z\}$ either commute or anticommute. Moving every $Q_i$ past the corresponding $P_i$ therefore gives
$$
PQ=(-1)^rQP,
$$
where $r$ is the number of positions containing distinct nonidentity Pauli matrices. Thus two Pauli strings always commute or anticommute. If they anticommute and a vector $|\psi\rangle$ were stabilized by both, then $PQ|\psi\rangle=|\psi\rangle$ and $QP|\psi\rangle=|\psi\rangle$, contradicting $PQ=-QP$. Their common <stabilizer subspace> is consequently the zero subspace $\{0\}$.
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