Stop Cisco AnyConnect from locking down the NIC
I believe you will need to setup split tunneling with a exclude list. So you don't tunnel your local lan traffic and everything else goes out the tunnel. This has to be configured on the ASA itself. Then in AnyConnect enable the option 'Enable local LAN access (if configured)'. You can enable it manually by clicking on the "preferences" button next to the "connect to" box or via XML profile.
Here is a link to the split tunneling information in the ASA 8.2 CLI guide http://www.cisco.com/en/US/docs/security/asa/asa82/configuration/guide/vpngrp.html#wp1053494