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Let $X$, $Y$ be complex algebraic varieties (possibly open and singular) and $A$ (resp. $B$) be a constructible sheaf of $\mathbb{Q}$-vector spaces on $X$ (resp. $Y$). Do we have the Kunneth type isomorphisms: $$H^k(X\times Y,A\boxtimes B)\simeq\bigoplus_{i+j=k}H^i(X,A)\otimes H^j(Y,B)?$$ Here $A\boxtimes B$ denotes $p_1^\ast A\otimes p_2^\ast B$.

P.S. If we consider the cohomology with compact supports, then we have $$H^k_{\rm cpt}(X\times Y,A\boxtimes B)\simeq\bigoplus_{i+j=k}H^i_{\rm cpt}(X,A)\otimes H^j_{\rm cpt}(Y,B)$$ following from the projection formula for pushforward with compact supports.

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Theorem 4.3.14 in Dimca's Sheaves and topology says this is true. The compact version, Corollary 2.3.31, is indeed much easier as you suggest. Another reference Dimca gives is Schuermann's Topology of Singular Spaces and Constructible Sheaves, Corollary 2.0.4 but I cannot check that at the moment.

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