Let $S$ be a noetherian scheme. Let $V,W$ be vector bundles on $S$. There is a canonical isomorphism of line bundles $$ {\rm det}(V\otimes W)\cong{\rm det}(V)^{\otimes{\rm rk}(W)}\otimes{\rm det}(W)^{\otimes{{\rm rk} V}}\,\,\,\,(\ast) $$ which is invariant under any base change. See eg

for this. There are similar identities for other tensor operations, eg $$ {\rm det}(\Lambda^2(W))\cong{\rm det}(W)^{\otimes({\rm rk}(W)-1)} $$

My question is: are there similar identities for perfect complexes in place of $V,W$ ?

Recall that an object in the derived category of ${\cal O}_S$-modules is called perfect if is Zariski locally isomorphic to a bounded complex of vector bundles.

One can show that if $V^\bullet$ and $W^\bullet$ are bounded complexes of vector bundles (on all of $S$) then there is an isomorphism $$ {\rm det}(V^\bullet\otimes W^\bullet)\cong{\rm det}(V^\bullet)^{\otimes{\rm rk}(W^\bullet)}\otimes{\rm det}(W^\bullet)^{\otimes{{\rm rk}(V^{\bullet})}}\,\,\,\,(\ast\ast) $$ where now ${\rm det}(\cdot)$ is the Knudsen-Mumford determinant of perfect complexes. This follows from identity $(\ast)$. However the isomorphism is not canonical. In other words, I don't know how to construct an isomorphism $(\ast\ast)$, which is functorial for isomorphisms in the derived category (or more concretely, for quasi-isomorphisms of bounded complexes of vector bundles on $S$). In particular, I don't know whether there is an isomorphism $(\ast\ast)$ (even a non canonical one) when $V^\bullet$ and $W^\bullet$ are only assumed to be perfect.

I would be grateful if anyone could share ideas, or direct me to references on this kind of problem. I am aware of Deligne's work on Picard categories and axiomatic descriptions of determinants but this seems to be of little help. One could try to prove an identity like $(\ast\ast)$ by showing that both sides satisfy the axiomatic properties of determinants (fixing $V^\bullet$ or $W^\bullet$) but such a verification seems difficult and tedious. Another way to proceed might be to write down an isomorphism $(\ast\ast)$ applying $(\ast)$ term by term and to verify functoriality for quasi-isomorphisms directly on the definition of the functoriality of the Knudsen-Mumford determinant but this again is difficult because this functoriality is defined in a very indirect way (see proof of Th. 1 in the paper of Knudsen-Mumford https://www.mscand.dk/article/view/11642 or

How to write down the determinant of a quasi-isomorphism?

). One would expect all the determinantal identities that are valid for vector bundles to be valid automatically for perfect complexes. There should be a way to show this.