TL;DR: You can localize entropy. For non-localized entropy, the symmetry is broken by the Von Neumann entropy of nondeterminate space (multiverse branching).

Let's look at things first macroscopically and then locally.

Macroscopic Entropy

Special relativity can be explained with clocks made out of oscillating light. Consider instead clocks made out of temperature disequilibrium e.g. ice melting into room temperature water.

Put one clock on a spaceship and launch it to Alpha Centauri and back. Leave the other one on Earth. Thanks to the twin paradox, the spaceship clock will melt less than the clock on Earth. But if space is orthogonal to change in entropy then what happened to the extra entropy the spaceship clock should have produced?

Just as in the twin paradox, there is a hidden asymmetry. In this case, the returning spaceship must have experienced a change in momentum. Something (whether reactant or a gravity assist) must have reversed the direction of the spaceship. This produces a change in momentum of something away from the Earth. The entropy gained from ejecting reactant at high speed far away from the Earth cancels out the entropy lost by the clock on the spaceship.

Conclusion: the macroscopic entropy of space is non-localized.

Local Entropy

Implicit in the concept of entropic time is that the time evolution components $\frac{1}{c^2}\frac{{\partial }^2}{\partial t^2}\psi +\frac{m^2{c}^2}{{\hslash }^2}\psi$ of the Klein–Gordon equation^[1] encapsulate a large localized change in entropy. This solves the problem of defining entropy locally but does not answer the question of how the integral of the change in entropy across space can be nonzero when space is defined as nonzero change in entropy.

The paradox is resolved by the nondeterminacy of space [LW · GW]. The entropy of space in the multiverse must factor in Von Neumann entropy $S=-{\sum }_j{\eta }_j\ln {\eta }_j$. Any particular decoherent spatial path can violate $\Delta S=0$ provided the total entropy change of the weighted probabilities of all possible decoherent states equals zero.

Conclusion: $\delta S=0$ applies to spatial movement in the multiverse, but not necessarily for a particular decoherent universe.

Putting them together

We have two big ideas:

1. Macroscopic measurements of entropy are non-localized.
2. Space is nondeterministic.

Macrostatic space-like paths with (seemingly) nonzero $\delta S$ hide their entropy via branching in the spatial multiverse. The multiverse's Von Neumann entropy cancels out the inhomogeneity of entropy in any particular decoherent space-like path.