I add an alternative answer.
If you add a moment to the hing but your are willing to accept suspended configuration, it will be a determinat structure.
Consider the hinge at lower level than the supports, then apply the moment between the beams in a way that it wants to open them.
Let L, be the length of each beam and W its weight. And the angle between the ...
If you think about it, a support is just an applied force or moment, right? If you have a simply supported beam with a force $F$ in the midspan, the supports will be two forces, each $F/2$ in the opposite direction.
But then, what if, instead of supports, you simply had two forces $F/2$ applied at the ends? That'd be exactly the same, right?
if you randomly put a moment at the center hinge, then the problem would still be indeterminate.
Make the moment large the hinge will close up.
Make the moment small the hinge will open up (if you consider weight).
You would need to put a rotational constraint at the hinge point to make it stable. (I am not certain is 'statically determinate' would be an ...