Or $2\pi$, or $3\pi$, or any integer multiple of $\pi$. Any odd multiple corresponds to -1 + 0i and any even multiple corresponds to 1 + 0i, aka -1 and 1.
"Phase of a real number" is a little bit of a misleading label. What is required here is an understanding of the complex plane and what "phase" means in terms of a DFT bin value.
Your question is equivalent to "For what values of arg(z) is z a real number?"
If that is meaningless to you, I suggest you start by reading two blog articles of mine:
The Exponential Nature of the Complex Unit Circle
And the newest:
Angle Addition Formulas from Euler's Formula
There are of course many other searches. Your terms should be "complex plane real values" for a start.
This is essential foundation material for a lot of DSP concepts.