What is a mathematical definition of the Maxwellian spacetime?

While this problem originated in physics, the question is purely mathematical. Because Maxwell's equations were not invariant under the Galilean transformation, Maxwell proposed his version of spacetime. It is as a 4D collection of points, such that:

1. Between any two points $p(t, x, y, z)$ and $q(t', x', y', z')$ there is a definite temporal interval $T(p, q) = t' − t$.

2. Between any two simultaneous points $p(t, x, y, z)$ and $q(t, x', y', z')$ there is a definite Euclidean distance

$$R(p,q) = \sqrt{(x' − x)^2 + (y' − y)^2 + (z' − z)^2}$$

3. Any worldline $\gamma$ through the point $p$ has a definite twist $\Omega(γ, p)$.

As a result of the third condition, linear acceleration is not absolute like in the Galilean spacetime. There is not enough structure in the Maxwellian spacetime to distinguish straight worldlines from curved worldlines.

However, rotation is still absolute, because the third condition allows telling when a worldline is "twisted". For a worldline $\gamma$ and a point $p$ on $\gamma$, the absolute rotation of $\gamma$ w.r.t. $p$ is given by $\Omega(γ, p)$.

(Image and partial content credit: Jonathan Bain, NY University.)

Properties of the Maxwellian spacetime:

1. No inertial frames (as opposed to many inertial frames in the Galilean spacetime).
2. Velocity is relative.
3. Acceleration is relative (as opposed to absolute in the Galilean spacetime).
4. Rotation is absolute.
5. Simultaneity is absolute.

The mathematical definition of the Galilean spacetime is a tuple $(\mathbb{R}^4,t_{ab},h^{ab},\nabla)$ where $t_{ab}$ (temporal metric) and $h^{ab}$ (spatial metric) are tensor fields and $\nabla$ is the coordinate derivative operator specifying the geodesic trajectories (Spacetime Structure).

A single metric does not work, because the speed of light is infinite, so time and space should be treated separately with the temporal metric:

$$t_{ab}=(\text{d}_a t)(\text{d}_b t)$$

and the spatial metric:

$$h^{ab}=\left(\dfrac{\partial}{\partial x}\right)^a\left(\dfrac{\partial}{\partial x}\right)^b+ \left(\dfrac{\partial}{\partial y}\right)^a\left(\dfrac{\partial}{\partial y}\right)^b+ \left(\dfrac{\partial}{\partial z}\right)^a\left(\dfrac{\partial}{\partial z}\right)^b$$

Finally, $\nabla$ on $\mathbb{R}^4$ is a unique flat derivative operator that for each coordinate $x^i$ satisfies:

$$\nabla_a\left(\dfrac{\partial}{\partial x^i}\right)^b=\mathbf{0}$$

In turn, the Newtonian spacetime is the same tuple with an additional structure $(\mathbb{R}^4,t_{ab},h^{ab},\nabla,\lambda^a)$ where $\lambda^a$ is a field that adds the preferred frame of rest:

$$\lambda^a=\left(\dfrac{\partial}{\partial t}\right)^a$$

What is a rigorous mathematical definition of the Maxwellian spacetime? Intuitively, it may be the same Galilean tuple, but with some additional structure (or a reduced structure) similar to how the Newtonian spacetime is created by adding $\lambda$ to the Galilean tuple. Thanks for your help!

Following your definition of a Galilean Spacetime, I will mostly rewrite what is found in https://arxiv.org/pdf/1707.02393.pdf. A Maxwellian spacetime is a tuple $(M,t_a,h^{ab},\Omega)$ where $M$ is a smooth manifold diffeomorphic to $\mathbb{R}^4$, $t_a$ is a temporal metric on $M$, $h^{ab}$ is a spatial metric on $M$, and $\Omega$ is a standard of rotation compatible with $t_a$ and $h^{ab}$. Further, the metric $h^{ab}$ is complete on spatial surfaces $\mathbb{R}^3_t$ foliated by $t:M\to \mathbb{R}$.

For any given $M$, we have a unique spatial derivative operator $D$ which obeys $D_a h^{bc}=0$.

(One change I am making here is not including the derivative in the tuple. It might be linquistics, but if there is a unique derivative $\nabla_a(\partial/\partial_i)^b=0$, we don't need to specify it in our choice of tuple. By picking $M$, we are automatically picking $\nabla$)