A Morse complex over $\mathbb{Z}$ is like a sheaf on a Legendrian front on $Spec(\mathbb{Z}) \times \mathbb{R}$ which is coherent along $Spec(\mathbb{Z})$ and constructible along $\mathbb{R}$.  Is that a hint for classifying such Morse complexes?  Any such Morse complex determines a (finite) collection of primes: those at which the ruling is over $\mathbb{F}_p$ is not the same as the ruling over $\mathbb{Q}$.  Is it easy to read these off the Morse complex?  What are the “local to global” obstructions — if I name any finite number of putative Morse complexes over various primes, plus the Morse complex over $\mathbb{Q}$, can I always glue? Probably it is best to first think about this question over the p-adics $\mathbb{Z}_p$.