Why the hydrogen radial wave function is real?
Is it a coincidence?
Wave functions which are Eigenfunctions of the stationary Schrödinger equation can always be chosen to be real. That's because the equation itself is real. Depending on the boundary conditions, the solution can also be complex (e.g. for scattering BC they are complex).
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Since quantum states that differ by multiplication by a complex number of length $1$ are all equivalent, you can multiply any wavefunction of the Hydrogen atom by such a complex number, and you'll get a vector in Hilbert space that is an equivalently valid description of the corresponding physical state.
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