How to obtain the correlation between monthly streamflows based on physical concepts?
One suggestion on generating the monthly flows is to use a seasonal ARMA model directly (e.g. see course notes), since you are looking at ARMA model for stochastic generation anyway. That would allow the explicit estimation of seasonal parameters for synthetic generation, and let you skip the step of going annual --> monthly, instead go monthly and calculate annual flows from the monthly series. You may have other reasons for wanting to start with annual flows to verify those first, but that is one suggestion.

In terms of physical concepts at a monthly flow resolution, this is harder to comment on without knowing what physical processes are impacting your streamflows, e.g. snowmelt, river ice and freezing, spring freshet, depression storage, etc. One suggestion may be to check monthly volumes and peaks as well as timing of peaks (such as spring freshet), and ensure that any synthetic time series match these statistics within some acceptable boundary, throwing out any simulations that are deemed unacceptable.

This is a pretty broad (research level) question but hopefully that gives some ideas.