Why is there no classical analogue to entanglement?

Quantum objects and classical waves have a lot in common, with many quantum effects having analogues in the world of classical waves. For instance, both have a notion of coherence, interference, and both evolve in accordance with a wave equation of some form (even if the Schrodinger equation is different from the usual wave equation with a second order time derivative). But (aside from measurement, but we won't get into that) there's one effect in QM that has absolutely no analogue whatsoever in the classical world (both in terms of linear waves and even in terms of the wide diversity of nonlinear waves): entanglement!

The usual explanation for why this is makes sense to me. Unlike classical waves, which can only exist in 3D space regardless of how many there are, quantum systems exhibit their wavelike behavior only in 3N dimensional space, where N is the number of interacting objects. So if you have N interacting classical waves in a system, you can describe them all as separate waves each evolving in 3D space via coupled PDEs, but if you have N quantum objects, this is no longer possible and you must describe the evolution of just a single wavefunction in 3N dimensional space. This single function in a higher dimensional space is capable of encoding correlations not possible if working just in 3D space, thus yielding entanglement.

My question is: why can't we mathematically describe N interacting classical waves as a single function evolving in a 3N dimensional space like we can with N interacting quantum objects? After all we can describe the evolution of an N body particle system via a single path in a 3N dimensional configuration space, so why can't we do this with waves? And even if we could, why wouldn't this evolution give us an analogue of entanglement? Thanks for the help!