Spotlight doesn't index internal drive when using an external startup drive

I've been running my late 2015 iMac 27" (bought new in 2017) from an external SSD startup drive for several months now.

The internal Fusion drive was simply too slow with the modern versions of MacOS.

Everything seems to work great except that....

I just noticed that when searching for files, either via Finder Search or Spotlight, that I don't get any results from the internal Fusion drive.

Also...

If I open a Finder Window, select the internal drive from the Side Bar, and limit a search to that internal drive, there are no results.

Is this a known problem and is there any solution or a workaround?

I don't use the internal drive for anything.

But I do make sure that its contents mirror my external startup drive via the SupeDuper! disk copying app just in case I ever do need to or want to startup from that disk.

So, if I see a search result for an item that's on my startup disk, it should also show a copy of that item on my internal drive too.

But it does not do that.

Any ideas?