How to write the gauge-invariant anomalous Nambu Green's function for 2D square lattice with uniform $pi$ flux?

For the free fermion system in two-dimensional square lattice, we add the $pi$ flux in each plateau:
$$H=-t sum_{langle i, jrangle} e^{i A_{i, j}} c_{i}^{dagger} c_{j}+h . c .$$
where
$$sum_{square} A_{i, j}=Phi=pi$$
and there exists gauge transform:
$$begin{array}{l}c_{i}^{dagger} rightarrow c_{i}^{dagger} e^{-i theta_{i}} c_{j} rightarrow c_{j} e^{i theta_{j}} A_{i, j} rightarrow A_{i, j}+theta_{i}-theta_{j}end{array}$$
I can obtain the dispersion by choosing a gauge, e.g.

Now, I want to write the gauge invariant Green’s function, some reference footnote 9 in page 68 says the gauge invariant Green’s function in the system with gauge field can be written as:
$$G(x)=-leftlangle c(x) c^{dagger}(0) e^{i int_{0}^{x} A d x}rightrangle$$
where $e^{i int_{0}^{x} A d x}$ is the straight line which connects beginning point and end point.

I can understand that the additional term $e^{i int_{0}^{x} A d x}$ recovers the gauge invariance of conventional Green’s function. However, I am confused that how to recovers the gauge invariance of BCS-type Nambu Green’s function, i.e. how to write the gauge invariant anomalous Green’s function like: $$F(x)=-leftlangle c(x) c(0) rightrangle$$