With the help of another developer here, I’ve resolved this.
The problem is that the uninstaller does not remove all 3840 DLLs from the GAC. Specifically, I found copies of iAnywhere.Data.SQLAnywhere.V4.0.dll and policy.12.0.iAnywhere.Data.SQLAnywhere.V4.0.dll and policy.12.0.iAnywhere.Data.SQLAnywhere.V4.5.dll in the C:\Windows\Microsoft.NET\assembly\GAC_MSIL folders for EBF 3840, even after I had uninstalled SQL Anywhere from my machine twice.
Once I deleted those DLLs, the problem stopped.