Background
I need to replace an existing certificate on a host named h1.vx.mydomain.net (not the real domain obviously).
The existing certificate says it was issued by a chain of CAs that I do not recognize. The CAs are clearly third-party CA organizations. The certificate was definitely not generated as a self-signed certificate. It also was not generated using an internal public key infrastructure.
What's puzzling to me is that the list of CAs in the certificate chain are not related to the Registrar for mydomain.net (it's registered with GoDaddy). I would have expected that the certificate chain would include GoDaddy as an Issuing CA or that one of the CAs in the chain would be a trusted intermediary CA under GoDaddy, but that doesn't seem to be the case.
This leaves me wondering what process to use to generate a replacement certificate for h1.vx.mydomain.net. I expected that I would need a GoDaddy certificate, but the old certificate makes me think GoDaddy was not used as the CA.
Main Question
Is it possible to have a "random" third-party CA generate a certificate using a CSR for a host in a domain not registered with the CA? If yes, is it going to have consequences?
I'm trying to make sense of a situation that seems foreign to me.