How to view thermal spacetime in "Minkowski" picture?

For things such as the Page-Hawking phase transition, we perform a Wick rotation, and consider the Free energy of the metric of a Black Hole in AdS (which has a periodic time to avoid conical singularities at the horizon), and compare it to the Free energy to "thermal AdS", which is the metric for the AdS vacuum in Euclidean coordinates, where the time coordinate is identified as $t_Eapprox t_E+frac{2pi}{T}$, giving gives our spacetime a temperature $T$ (through the usual picture of computation of correlations functions with the path integral).

My question is, how would we differentiate the regular AdS vacuum from thermal AdS in the non-wick rotated picture, i.e. with real time ? It would appear that the condition $t_Eapprox t_E+frac{2pi}{T}$ translates to some imaginary periodicity of the real time, something like $tapprox t+frac{2pi i}{T}$. I really don’t know how this makes any sense, and in turn I am unable to understand physically what is "thermal AdS", since the physical spacetime is living in real coordinates, and not in Wick-rotated imaginary coordinates.

So, how do we view "thermal AdS" in real coordinates, and how does it relate to the time-periodicity of the euclidean time coordinate ?