Here’s the soultion I’ve found. It’s clumsy but seems to work.
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Add the Variant
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Click “Save”
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An error message will show up. Ignore it and …
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Reload the page.
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Each variant will now have SKUs that, apparently, the system automatically assigns. Or at least some of the variants will now have SKUs. I noticed it sometimes take muliple reloads for them all to populate. But if you have your own SKUs you can skip additional reloads and …
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Click on an individual product/variant. This will open a page that lists all the variants. From here you can add you own SKU’s for each variant if needed, one at a time. Once you’re done …
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Click the back button and return to the product page. Click Save. It should now save without an error message.
If you just want to use the SKUs that Shopify’s system (apparently) populates on its own, you may try skipping steps 6 and just reloading a few times, then once they all have SKUs, hitting “Save.” I haven’t tried that approach myself.