I have with me the non-archimedean local field of characteristic $p,\ K = \mathbb{F}_p((t))$, and it's maximal unramified extension $K^{\text{ur}}=\overline{\mathbb{F}}_p((t))$; letting $p\neq2$, I then also have the unique totally tamely ramified extension of $K^{\text{ur}},\ K^{\text{ur}}(t^{1/2}):=\overline{\mathbb{F}}_p((t^{1/2}))$.
Now, I have a smooth projective variety $X$ and its Symmetric Square $S^2X = (X \times X)/S_2$, and it's diagonal $\Delta$. I have to show that $$\left(S^2 X \setminus \Delta\right)(K^{\text{ur}}) = S^2 X (K^{\text{ur}}) \cup X\left(K^{\text{ur}}\left(t^{1/2}\right)\right)/(\mathbb{Z}/2\mathbb{Z})$$
I'm just at my wit's end on how to show this. Can I get any help?
I suppose I understand how the second term in the union will be mapped to the left hand side, as 2-tuple with an element and its Galois-conjugate i.e. $(a,a^{\sigma}) \sim (a^{\sigma},a)$