Can a system have constant phase response other than +/- 90 deg (and multiples)?

I have seen frequency responses where amplitude response is constant( pure delay) and where phase response is constant(pure integrator).

In an integrator the constant phase is -90 deg. In a pure gain block the constant phase is 0 deg. Can we have a system which has a constant phase of say -45 deg?

I tried to find such a simple system analytically and ended up with an expression of system function as $$H(j\omega)=e^{-j\pi/4}$$ Can we have a such a system in real life?

Background: I am trying to understand the importance of phase and amplitude plots individually. For eg: In the frequency response of pure delay systems, amplitude plot is constant and phase plot is a linear function of frequency. Similarly how will the system behave if

1) the phase is constant(other than -90 deg) and amplitude plot is a linearly decreasing function of frequency?

2) the phase plot is constant (say -90 deg, -45 deg)) and amplitude plot is also constant

If you're talking about ideal conditions, then, sure, you can have exotic transfer functions with $$\\tanh{s^\frac{2}{3}}\$$, or what have you. I don't know where this one would be useful, but sure, you can. 