I've been having this problem too. I'm not sure this would work for every circumstance but I found a solution for my particular case.
Mac OS X Sierra using an iPhone 6s for tethering. Anyconnect 4.3.02, ASA 5506-X 9.6(3)
This article is very helpful in explaining some of the complexity.
I only wanted DNS to be public, but apparently there is a bug (described above) that doesn't allow that. So you have to send a public DNS server to your client. If you don't do this you will find your /etc/resolv.conf empty when the VPN is connected. If you do set a public DNS server like 18.104.22.168 it will automatically include the secure route to your client so you can tunnel it. There is an option to split tunnel and not send traffic for 22.214.171.124 through the tunnel but that apparently doesn't work on these versions of code apple/cisco. So I allowed the DNS traffic through the tunnel but in my case initially was dropping it due to misconfigured NAT rules. Once that was fixed I am able to resolve DNS names using a public DNS server although the tunnel. If you had DNS at your site you could put your private DNS server in place of 126.96.36.199 and enable recursion I suppose. Hope this helps someone.
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