A tab keeps the organization it is working in
Switch organizations in one browser tab and the others stay where they are, each still acting on the organization it is showing. A second tab can be left open on another organization for as long as you like.
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Only half of the switch used to be per tab. What a tab displayed and which roles it gave you stayed with the organization you had opened it in, while everything it sent to the server quietly moved to the newly chosen one — so a tab could list one organization’s projects and create the next one somewhere else, and an action your role allowed here was refused as though it did not. Which organization each tab is in is now settled per tab and survives a reload; a newly opened tab still starts in the one you chose last.