I am trying to understand the physical meaning of phase in the order parameter of a superconductor. In particular,I was looking at this article that states the phase $e^{i\theta}$ in the BCS ground state

$$|\psi_G\rangle=\prod_k (u_k +v_k e^{i\theta}c_{k\uparrow}^\dagger c_{-k\downarrow}^\dagger)|\phi_0\rangle $$

is not a gauge invariance, but a global $U(1)$ phase rotation symmetry. However, the author (on page 6) then says that

The most significant difference between an antiferromagnet and a superfluid or superconductor with regard to the order parameter is that the broken rotational symmetry in the former case is much more evident to us, as all the macroscopic objects in our daily life experience violate rotational symmetry at one level or another...In the case a superconductor, we need a second superconductor to have a reference direction for the phase, and an interaction between the order parameter in both superconductors to detect a relative difference in the phases.

I can easily visualize this breaking of $U(1)$ symmetry in a antiferromagnet. However, I am still confused as to what, exactly, is the physical meaning of phase in the superconducting order parameter? In particular, how can I prescribe a physical meaning to the phase on the Cooper pairing term in the BCS ground state? As with my other post on a similar question I had, any explanations or resources at the level of Tinkham would be greatly appreciated.

I am trying to understand the physical meaning of phase in the order parameter of a superconductor. In particular,I was looking at this article that states the phase $e^{i\theta}$ in the BCS ground state

$$|\psi_G\rangle=\prod_k (u_k +v_k e^{i\theta}c_{k\uparrow}^\dagger c_{-k\downarrow}^\dagger)|\phi_0\rangle $$

is not a gauge invariance, but a global $U(1)$ phase rotation symmetry. However, the author (on page 6) then says that

The most significant difference between an antiferromagnet and a superfluid or superconductor with regard to the order parameter is that the broken rotational symmetry in the former case is much more evident to us, as all the macroscopic objects in our daily life experience violate rotational symmetry at one level or another...In the case a superconductor, we need a second superconductor to have a reference direction for the phase, and an interaction between the order parameter in both superconductors to detect a relative difference in the phases.

I can easily visualize this breaking of $U(1)$ symmetry in a antiferromagnet. However, I am still confused as to what, exactly, is the physical meaning of phase in the superconducting order parameter? In particular, how can I prescribe a physical meaning to the phase on the Cooper pairing term in the BCS ground state? As with my other post on a similar question I had, any explanations or resources at the level of Tinkham would be greatly appreciated.