"This Speaker Can Only Be Controlled by People Sharing a Home in the Home App" Issue on HomePod Mini

Are you using a guest wi-fi setup that blocks peer to peer traffic, or the feature that gives you a private network address? You don't want either on you own local Wi-Fi as they might interfere with the way that HomePod validates connections. Try switching to Anyone On the Same Network if it still doesn't work after checking on those two. See Play audio on HomePod using an iOS or iPadOS device - Apple Support.

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