1) We don't have a control on source syste, we can only perform query on itI was referring to a materialized view on the system where Apex is installed, not on the source database.
2) There are many tables involvedI don't understand why this is a problem. Too much data I can see, but too many tables... not so much.
3) Data has to be in real time, with no delayThis would a problem for MV or collections. The collections would store the data as of the initial query. Any IRs using the collection after the fact would be using stale data. If you absolutely have to have the data as of right now every time, then the full query must run on the remote system every time. Tuning that query is the only option to make it faster.
4) There are many transactions on the source tables (they are the core of the source system) and so MV could not be refreshed so fastProbably could be with fast refresh enabled, but not necessarily practical to do so. As I indicated in 3, you have painted yourself into a corner here. You have indicated a need for a real-time query and that eliminates a number of possibilities for query-once use-many performance solutions.