  We provide a complete classification of the algebraicity of (generalized)
hypergeometric functions with no restriction on the set of their parameters.
Our characterization relies on the interlacing criteria of Christol (1987) and
Beukers-Heckman (1989) for globally bounded and algebraic hypergeometric
functions, however in a more general setting which allows arbitrary complex
parameters with possibly integral differences. We also showcase the adapted
criterion on a variety of different examples.
