What differs between post-filter and global aggregation for faceted search?
In both cases Elasticsearch will end up doing mostly the same thing. If I had to choose, I think I'd use the global aggregation, which might save you some overhead from having to feed two Lucene collectors at once.
The actual solution we used, while not a direct answer to the question, is basically "neither".
From this elastic blogpost we got the initial hint:
Occasionally, I see an over-complicated search where the goal is to do as much as possible in as few search requests as possible. These tend to have filters as late as possible, completely in contrary to the advise in Filter First. Do not be afraid to use multiple search requests to satisfy your information need. The multi-search API lets you send a batch of search requests.
Do not shoehorn everything into a single search request.
And that is basically what we are doing in above query: a big bunch of aggregations and some filtering.
Having them run in parallel proved to be much and much quicker. Have a look at the multi-search API