I'm learning about Lie groupoids and was inspired (by Mackenzie's book) to consider the following problem.

Consider first a principal bundle $P\xrightarrow G M$; we can construct the quotient manifold
$$
\Omega=\frac{P\times P}{G},
$$
obtained as the orbit space of the pair action $g(u_2,u_1)=(gu_2,gu_1)$. In this way, we obtain a Lie groupoid (called the *gauge groupoid* and denoted $\Omega\rightrightarrows M$) with the source and target maps $s,t\colon \Omega\rightarrow M$ given by $s[u_2,u_1]=u_1$ and $t[u_2,u_1]=u_2$, and partial multiplication $[u_3,u_2'][u_2,u_1]=[u_3,u_1]$ iff $u_2'=u_2$.

In particular, the universal cover $\mathrm{SU}(2)\rightarrow \mathrm{SO}(3)$, given by the adjoint representation, may be observed as a principal bundle ${\mathrm{SU}(2)}\xrightarrow{\mathbb Z_2}{\mathrm{SO}(3)}$. The map $\mathrm{SU}(2)\times \mathrm{SU}(2)\rightarrow \mathrm{SO}(4)$, given by $(p,q)\mapsto (x\mapsto pxq^{-1})$, where $p,q,x$ are viewed as quaternions, induces the isomorphism of Lie groups $$ \frac{\mathrm{SU}(2)\times \mathrm{SU}(2)}{\mathbb Z_2}\cong \mathrm{SO}(4), $$ so we actually obtain a Lie groupoid $\mathrm{SO}(4)\rightrightarrows \mathrm{SO}(3)$, with the source and target fibres diffeomorphic to $\mathrm{SU}(2)$. Additionally, for any $x\in \mathrm{SO}(3)$, the vertex group $s^{-1}(x)\cap t^{-1}(x)\cong\mathbb Z_2$, indicating that $\mathrm{SO}(4)\rightrightarrows \mathrm{SO}(3)$ should be isomorphic to the fundamental groupoid of $\mathrm{SO}(3)$.

Now consider the (physically more juicy) universal cover $\mathrm{SL}(2,\mathbb C)\xrightarrow{\mathbb Z_2}{\mathrm{SO}^+(1,3)}$, where $\mathrm{SO}^+(1,3)$ denotes the component of the Lorentz group $\mathrm O(1,3)$ which contains the identity matrix -- this is a Lie subgroup (the proper ortochronous Lorentz group). By the general construction, we obtain a gauge groupoid $$ \frac{\mathrm{SL}(2,\mathbb C)\times \mathrm{SL}(2,\mathbb C)}{\mathbb Z_2}\rightrightarrows \mathrm{SO}^+(1,3). $$ We would again like to identify the space of arrows with a known space, but it is now 12-dimensional. The only sensible candidate that comes to my mind is the complexification $\mathrm{SO}^+(1,3)_{\mathbb C}$, but I don't know where to start confirming whether this is the case. So the question is: $$ \frac{\mathrm{SL}(2,\mathbb C)\times \mathrm{SL}(2,\mathbb C)}{\mathbb Z_2}\stackrel{?}\cong\mathrm{SO}^+(1,3)_{\mathbb C}. $$

Lastly, if anyone has any intuition regarding the obtained gauge groupoid over $\mathrm{SO}^+(1,3)$, feel free to make a hand-wavey explanation of what this object could physically represent.