This is an undocumented problem with Cisco AnyConnect.
Symptoms:
You can connect with Cisco AnyConnect and you pull an IP address from either your DHCP Server or yor Local IP Group on the ASA. If you go into the details of the connection, the split tunneling is working correctly. Yet, when you try to access a shared network folder or application, you are unable to. If you try to access a network share, you are prompted for your username and password. It's as if your Windows Logon credentials aren't being passed along for authentication.
How to Resolve:
For some unknown reason, to solve this issue, you need to set the WINS server values in your group policy on the ASA for the connection. If you are like me, you don't have a WINS server setup in your environment, so just use your DNS servers:
ASA>enable
ASA#conf t
ASA(config)#group-policy <YOUR POLICY> attributes
ASA(config-group-policy)#wins-server value x.x.x.x