Why do people use real values for the Wick-rotated time $tau$?

In doing instanton problems or when connecting quantum field theory to statistical mechanics, I often see people trying the Wick rotation trick by defining an imaginary time $tauequiv it$. So, in this case, we should make sure that $tau$ is imaginary, right? However, I often see places where people seem to handle $tau$ as a real number, for example setting $tau$ to $pminfty$. Moreover, I remember in one of my quantum mechanics classes, the professor said the Euclidean time $tau$ was real. I am so confused and need some good explanations.