Sorting Eigensystem According to Complicated Rule

I have looked for an answer to this but the near duplicates I could find seemed slightly distinct.

I have a matrix $A$ which has eigenvalues in pairs $lambda_1,-lambda_1,lambda_2,-lambda_2,dots$. I would like to sort the eigensystem such that the eigenvectors are in this order, with the eigenvalues having descending real parts. That is, I want to sort in descending order of the function $f=|Re(cdot)|$ and break ties by $g=Re(cdot)$.

What I was hoping for was something like:

f[z_] := Abs[Re[z]];
g[z_] := Re[z];
{eval,evec} = SortBy[Eigensystem[N[A]][Transpose],{f,z}][Transpose];

but this doesn’t work. Replacing {f,g} with Abs@*Re does work but not for the tiebreak (neither does {Abs@*Re,Re}).

An example matrix that is $8times 8$ is the following (although any anti-symmetric matrix has plus/minus pairs of eigenvalues):

A = {{0, 0, -1, -4*I, 0, 0, 0, 0}, 
    {0, 0, 0, -1, 0, 0, 0, 0}, {1, 0, 0, 0, -1, -4*I, 
      0, 0}, {4*I, 1, 0, 0, 0, -1, 0, 0}, 
    {0, 0, 1, 0, 0, 0, -1, -4*I}, {0, 0, 4*I, 1, 0, 0, 
      0, -1}, {0, 0, 0, 0, 1, 0, 0, 0}, 
    {0, 0, 0, 0, 4*I, 1, 0, 0}}

Edit:
Additionally, I have realised since writing this that for my application I probably want to organise the sorting differently than I’ve asked for. In particular I want to order first by f[z_] :=Abs[Re[z]], then break ties first by g[z_]:=Re[z]*Im[z] and second by h[z_]:=Re[z]. So a solution that enables me to simply list three functions f,g,h which I’ve defined elsewhere in my code would be preferable.