A Banach space operator $S$ satisfies property $(aw)$ if $\s(S)\setminus\sw(S)=E_a^0(S)$, where $E_a^0(S)$ is the set of all isolated point in the approximate point spectrum which are eigenvalues of finite multiplicity. Property $(aw)$ does not transfer from operators $A$ and $B$ to their tensor product $A\otimes B$, so we give necessary and/or sufficient conditions ensuring the passage of property $(aw)$ from $A$ and $B$ to $A\otimes B$. Perturbations by Riesz operators are considered.