> the math works, and it looks like it should be possible

The math can only work (if it even does) in exceptionally constrained circumstances.
The main purely-theoretical issue is that the paper's <https://arxiv.org/abs/2102.06824v2> solution is wholly spherically symmetric, and with vacuum everywhere but in the warping region.

"Fig. 1: Asymptotically-flat vacuum background ... general stationary curved region with a spherical topology ... (the warping region) ... flat inner region ... ('passenger' space). ... As we discuss in Section 5, warp drive spacetimes require some form of propulsion in order to accelerate. For this reason, in physical realisations of such spacetimes, the front and rear parts are likely asymmetric". [emphasis mine]

Do the maths fall apart if we deviate from total spherical symmetry, either by changing the front and rear parts, or by breaking the vaccuum condition on the inside and outside of the shell, especially if our matter is not precisely spherically distributed?

They do a review of this, for limited deviations from spherical symmetry, in §4. However, the vaccum condition is what they study, not straying very far from their earlier "if [the warp region] were replaced by Minkowski spacetime, the whole spacetime would be Minkowski space."

The paper declines to answer the question about the distribution of matter (i.e., not-vacuum): "all warp drive spacetimes are asymptotically-flat ... no metric which describes an accelerating warp drive solution has been presented in the literature ... metrics for ... accelerating objects and more general axisymmetric objects which preserve their shape and mass ... remains a subject for future studies". That last part suggests that the inside part might be kept appropriately symmetrical through dynamic reconfiguration of the things inside (i.e., the ship, cargo, fuel, passengers).

But what about outside the warping region? In our solar system, and in our galaxy, we are not in Minkowski space.

The outer part of the warp region, being asymptotically flat, might be amenable to a "thin shell" (Israel/Darmois junction formalism) with the warp bubble far enough away from everything outside that it feels no tidal forces, which for the purposes of the text above is "acceleration". What determines how far is far enough away? The authors don't attempt an answer.

There are plenty of things (stars, most notably) in our part of the galaxy that will induce tides. The solar component of the lunisolar tide in Earth's oceans is substantial at 1 astronomical unit from our sun. That solar component doesn't go away when a space craft is (for example) 2 a.u. away from Earth&Moon on the opposite side of the sun. Can one zip this warp bubble safely past a 1 solar mass star at a distance of 1 au? Ten au? Can the warping region or passenger region be adjusted to allow for close approaches/operation within a star system? All questions for some possible future paper.

Apart from stars, we inhabit a region of space which is dotted with things you can smack into (from the very solid like rocks and iron to the very wispy, stuff will still tend to slow you down when you hit it; deceleration and acceleration are the same issue). Near luminous enough stars, radiation pressure might also induce non-negligible accelerations. Is any of that relevant? If relevant, is it "fixable"? How? Alter the shape of the warping region, or the passenger region, or both?
Questions for some possible future paper.

So, even if one accepts for the sake of argument that the maths are flawless and further one just assumes the existence of controllable negative energy densities, this paper's maths are designed for a spacetime which overall isn't very much like what we have within a few hundred lightyears of here. Maybe some future paper will attempt to show that a warp bubble can work in more realistic spacetimes than almost-everywhere-empty-of-matter-and-feeling-no-gravitation setting of this paper. ("Almost-everywhere" means that there is matter and curvature in the warping region, and maybe the passenger region, but nowhere else,
not even a destination star a dozen light-years away).