Quite a few VPNs (like Sonicwall NetExtender) will enforce the use of a corporate-enforced DNS. I have that issue with one of my customers that requires me to use their VPN to remotely manage their servers, and they push 9.9.9.9 as DNS, preventing me from even doing local name resolution while the VPN is up.
Unfortunately this is a DNS issue, split tunnelling cannot help here (I do have split tunnelling set up with that VPN).