Hello,
So the split tunnel already allows the route to behave how you want it unless you meant tunnelall.
split tunnel = only traffic headed to company network will be sent through vpn connection --- other traffic will go out through normal network connection.
tunnelall = default config, all traffic from your computer will go through VPN connection.
If your ASA is already setup for split tunneling, I believe you probably need to verify network settings on your PC. Verify that "Use default gateway on remote box" is unchecked under the Network Connections -> Your VPN Connection (Properties) -> Networking -> IPv4 -> Properties -> Advanced tab. Depending on what version of Windows you have, you may have to disable it differently.