Little-group and spinor helicity

I’m struggling a bit to understand the little group in the context of massless momenta and the spinor-helicity formalism.

I’ll clarify notation and my understanding through a brief recap, and put the questions at the end.

We can express a momentum vector $p^mu$ as a matrix through
$$
p^{alpha dot{beta}}
=
p^mu sigma_mu^{alpha dot{beta}}
=
begin{pmatrix}
p^0 – p^3
& -p^1 + i p^2

-p^1 – i p^2 & p^0 + p^3
end{pmatrix}
$$
which for lightlike $p$ is a rank-one matrix, so can be expressed as an outer product of two spinors,
begin{align}
|prangle^{alpha}
&=
frac{t}{sqrt{p^0 – p^3}}
begin{pmatrix}
p^0 – p^3

-p^1 – i p^2
end{pmatrix}

[ p |^{dot{beta}}
&=
frac{t^{-1}}{sqrt{p^0 – p^3}}
begin{pmatrix}
p^0 – p^3 & -p^1 + i p^2
end{pmatrix},
end{align}
for any choice of $t$. Then from the Dirac equation(?) we impose
$$
left(|prangle^{alpha}right)^ast
=
[ p |^{dot{alpha}},
$$
which imposes $t in mathbf{R}$.

The (Wigner) little group is the subgroup of Poincaré (?) transformations that fixes $p$ (the stabiliser of $p$?), which can be seen most easily if we boost into the momentum-aligned frame
$$
p^mu = begin{pmatrix}
{E} & {0} & {0} &{E}
end{pmatrix}.
$$
Any rotation around the new $z$-axis, or any translation in the $xy$-plane, will leave $p^mu$ invariant.

We can use spherical coordinates for $mathbf{p}$ to parametrise a massless momentum vector as
$$
p^mu = E begin{pmatrix}
{1} & {sin theta cos phi} & {sin theta sin phi} &{cos theta}
end{pmatrix}.
$$
Then
$$
p^{alpha dot{beta}}
=
2 E
begin{pmatrix}
sin^2 frac{theta}{2}
& – e^{-iphi} sin frac{theta}{2} cos frac{theta}{2}

-e^{iphi} sin frac{theta}{2} cos frac{theta}{2}
&
cos^2 frac{theta}{2}
end{pmatrix}
$$
which can be factorised as
begin{align}
|prangle^{alpha}
&=
t
sqrt{2E}
begin{pmatrix}
sin frac{theta}{2}

-e^{iphi} cos frac{theta}{2}
end{pmatrix}

[ p |^{dot{beta}}
&=
{t^{-1}}{sqrt{2E}}
begin{pmatrix}
sin frac{theta}{2}
&
-e^{-iphi} cos frac{theta}{2}
end{pmatrix},
end{align}
again for any choice of $t$. (this factorisation is equivalent to the above up to a redefinition of $t$ to incorporate a factor of $mathrm{sgn} left[ sin frac{theta}{2} right]$ arising from the square root)

Clearly this coincides with the momentum-aligned frame for $theta = 0$, and there is then a symmetry corresponding to the (now redundant) choice of $phi$. Is this redundancy essentially the little group?

For $theta = 0$ we then have
begin{align}
|prangle^{alpha}
&=
-e^{iphi}
t
sqrt{2E}
begin{pmatrix}
0

1
end{pmatrix}

[ p |^{dot{beta}}
&=
-e^{-iphi}
{t^{-1}}{sqrt{2E}}
begin{pmatrix}
0
& 1
end{pmatrix}.
end{align}

So in this case $t$ and the exponential factor could be combined into a single factor, if we wanted to. The $t$ factor alone is often (e.g. in Elvang and Huang) described as the ‘effect of the little group’ on the angle and square spinors. Is this conflation with $e^{iphi}$ the reason for this? Why does it hold in any other frame than this one (if it does)?

Finally if we move to complex momenta, we can use the same parametrisation with complex $E, theta, phi$ and everything should carry through, with
$$
e^{iphi}
=
e^{- Im [phi]} left( cos Re [phi] + i sin Re [phi] right)
.
$$
What is the little-group for complex kinematics isomorphic to and how is this manifest here?