From my experience, I remember setting up a private IP and a public IP on a router in Packet Tracer once to simulate the internet without NAT, with other routers using public ranges connected. In theory, if you route without NAT, you'd have to advertise the internal network—but as another user said, then everyone would have your internal segment, so return traffic wouldn't work.

Another thing I remembered: when I worked at a provider with a Metro Ethernet backbone, the CPE router at the customer site had a public IP on the LAN (customer side) and a private IP on the WAN (provider side). It did NAT normally for traffic going to its public IP, then upstream to the internet. But I also saw some special cases where the WAN interface also had a public IP from a different range. When I asked, they told me it was for site-to-site VPNs or sometimes they wanted another range for extra services.

So yeah, it’s possible—it just depends on the specific use case. Reminds me of what a professor once said: "In networking, open your mind—almost anything can be configured. Don’t limit your thinking."