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I believe you can do so by renaming the stack - see <https://www.pulumi.com/docs/reference/cli/pulumi_stack_rename/>

I think if you had code to create identical resources you could export a monolithic stack to a file, shard it out, and then import the resulting files into new stacks - see <https://www.pulumi.com/docs/reference/cli/pulumi_stack/> import/export commands

thanks for that!  I actually see in the UI now for stack settings there’s a “transfer stack” option to move a stack to another organization, so that answers that one

splitting up an existing stack via that export/import process sounds a bit of a pain

would imagine that must be a common thing that people do

No problem. I can't speak for other people, but personally I haven't split up a stack before. That's just where I'd start looking if I was going to.