The complete identification problem is to decide whether a stochastic process (Xt) is a hidden Markov process and if yes to infer a corresponding parametrization. So far only partial answers to either the decision or the inference part have been given all of which depend on further assumptions on the processes. Here we present a full, general solution for binary-valued hidden Markov processes. Our approach is rooted in algebraic statistics hence geometric in nature. We demonstrate that the algebraic varieties which describe the probability distributions associated with binary-valued hidden Markov processes are zero sets of determinantal equations which draws a connection to well-studied objects from algebra. As a consequence, our solution provides immediate algorithmic access where tests come in form of elementary (linear) algebraic routines. AMS 2000 subject classifications: Primary 62M05, 62M99; secondary 14Q99, 68W30.