I’m thinking that this is likely similar to what we saw in the Flynn case. I imagine the “Alterate Writ” is perhaps incorrect nomenclature - IANAL - so take it all with a large grain of salt.
What we saw in Flynn was the Appellate court demanding justification from Sullivan. This isn’t the Writ of Mandamus itself, but rather a part of the upper court decision making process. They are developing a record on the issue by getting responses from the court in question.
If the justification doesn’t hold water - then the upper court issues the actual “Writ of Mandamus.”
Again - IANAL - but this seems to be similar and a likely explanation.
Prior to the Flynn case, I don't ever recall hearing of a "writ of mandamus" from a higher court to a lower court. I think that is because the lower court is required by law to defer to the higher court and every order from the higher court has the same weight as a writ of mandamus.
My prior understanding of the purpose of a writ of mandamus was as a way for any court to make a non-court party do a particular thing. Sort of the opposite of an injunction, which is a court order which bars a party from doing a particular thing.
I'm guessing that writs of mandamus are rare because judges have alternatives that don't require the cooperation of a non-court party. Rather than mandate that a business surrender certain property, for example, the court can authorize law enforcement to seize the property. It is better to involve law enforcement rather than tempt the businessman who has a conflict of interest.