Systematically outputting sign vectors of restricted hyperplane arrangement

[Will Dana]

(I have already tried this question on ask.sagemath.org, but did not get any response; I figured it was worth trying here.)

I am working on CoCalc, in Firefox.

I have a way of constructing hyperplane arrangements, which involves taking the restriction of a larger arrangement to one of the hyperplanes in it. I want to output a list of the sign vectors of the regions of this arrangement, where the order of the coordinates in the sign vector corresponds to a specific order I want to impose on the hyperplanes. This has turned out surprisingly tricky, in part because the hyperplane arrangement constructor seems not to preserve the order of the hyperplanes fed into it. For example,

H = HyperplaneArrangements(QQ, ('x0','x1','x2','x3'))
H([0,0,0,0,1], [0,1,0,1,0], [0,0,1,1,0])

produces the output

Arrangement <x3 | x1 + x2 | x0 + x2>

with the latter two swapped.

Thus if I run the built-in sign_vector method, the coordinates of the resulting hyperplanes may be arbitrarily rearranged from how I want them.

I could use the equations for my hyperplanes to evaluate sign vectors manually, except that I'm working in a subspace (since I restricted to a hyperplane) and I'm also not sure how to systematically adapt my equations to how Sage coordinatizes this subspace.

Does anyone have advice on how to resolve either of these arbitrary decisions, or how to find a different workaround?