So it's been 2 years since ubunonymount attempted to solve the problem I still face, today (2019):
Acronis True Image, running in a Windows 7 VMware Workstation virtual machine, will not accept \\vmware-host\Shared Folders\TIBackupDestination as a "Back up to:" destination. If I either attempt to browse to \\vmware-host\Shared Folders\ or explicitly enter the full UNC path to the target folder, an Acronis "Authentication Settings" dialog demands a user name and password for folder access.
I've specified the userid and password that owns the folder on the VMware host, the userid and password under which the user account can read and write to it from within the VMware guest VM (same credentials, in my case). I've even granted "Full Control" to "Everyone" on the directory in the host machine's OS.
Mapping a network drive in the guest VM doesn't do any good (I've read in other posts that mapped drives aren't supported by other Acronis products, so I'll bet Acronis True Image doesn't support them, either).
Creating a symbolic link in the guest VM to the UNC path -- a trick I learned from an Acronis forum post at [SOLVED] Accessing .TIB from vmware-host VM share folder | Acronis Forum -- (mklink /D C:\WriteBackupHere \\vmware-host\Shared Folders\TIBackupDestination) doesn't work (Acronis True Image doesn't even "see" the symbolic directory).
I'm out of ammo.
Anybody else have a solution?