An observation is that the session ID in the URL of application A changes(on refresh) when application B in another tab is opened.
are Application A + B in the same Apex Workspace?
If YES, then you can set
Application -> Shared Components -> Authentication Schemes -> Edit current Schema
"Cookie Name" to one common string in both applications (case sensitive!).
If you start Application A now and call Application B in a second step with the Session_id from Application A, they should both co-exist.
If this works for you, have a look at
for how to build a simple SSO solution.
A completly different approach is the new feature APEX_CLONE_SESSION in APEX 5.1