Confirming the VPN cause: it applies on the PC side too. Full debugging from my side below.
My setup: iCloud for Windows 15.10.39.0 (Microsoft Store), Windows 11 26H2 (26300.9711). Same symptoms as in this question: the password is accepted, but sign-in or "Verify your Apple Account" fails with "Authentication failed. Retry will be attempted.", "iCloud for Windows has not fully initialized", or "error connecting to the Apple Account server". No approval prompt ever shows up on the iPhone.
In the iCloud logs (%LOCALAPPDATA%\Packages\AppleInc.iCloud_nzyj5cx40ttqa\LocalCache\Local\Logs\iCloudHome.*.log) I saw:
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AKValidateAppleIDCredentials returned -9106 (also -7005)
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piggybackErrorCode=-9106, statName "approveDevice"
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Failed to obtain public APS push token / Device registration failed
The fix: disconnect Tailscale from the tray icon, click Verify again, approve on the iPhone, then reconnect. I wasn't using an exit node. Just having Tailscale "Connected" (with MagicDNS on) was enough to break the device-approval step. My iPhone is also on the same tailnet, which may matter. Once the PC is verified, everything keeps working with Tailscale back on.
So whatever VPN you use (Tailscale, WireGuard, corporate VPN, etc.), turn it off on both the PC and the iPhone for the sign-in, as the OP found with the VPN on the iPhone. Things I tried first that did not help: rolling back 26H2, resetting and reinstalling iCloud, wiping Apple registry/data, and removing third-party smart-card middleware.
Two other things that also bit me along the way:
- Windows Firewall Control (or any outbound-blocking firewall) silently blocks the iCloud executables. Allow everything under
C:\Program Files\WindowsApps\AppleInc.iCloud_*\iCloud\, plus msedgewebview2.exe for the verification window.
- Advanced Data Protection adds an extra device-approval step, which is what kept failing for me. Turning it off temporarily changed the error, but the real fix was Tailscale.
Full write-up on Apple Community: https://discussions.apple.com/thread/256383864