>> Largely backwards compatible

> how exactly in the case of IP?

For example, introduce two special IP options that would act as "page selector" for source and destination addresses respectively. i.e.

  SRC: 1.1.1.1, DST: 2.2.2.2, no options ->
      from 1.1.1.1 to 2.2.2.2
      (backwards-compatible packet, classic IPv4)

  SRC: 1.1.1.1, DST: 2.2.2.2, opt.X: 1.1.1, opt.Y: 2.2.2 ->
      from 1.1.1.1.1.1.1 to 2.2.2.2.2.2.2
      (our faux-IPv6)

  SRC: 1.1.1.1, DST: 2.2.2.2, opt.X: 0.0.0, opt.Y: 2.2.2 ->
      from 1.1.1.1 to 2.2.2.2.2.2.2
      ("old" address communicating with "new" address)

Last example of three is impossible with IPv4/6 dichotomy, and is most important for backward compatibility here - yes, "old" IP holder needs to upgrade their software too[1], but they don't need to get a new IP address, and as a result you don't end up with two isolated "Internets", a.k.a. dual-stack.

Say, for ease of processing on router, require that X and Y would appear only together, in that order, preceding other options (which is a bit of an exception as far as IP options go, but again, a backward-compatible one).

Now, I'm not proposing this seriously of course, but you asked for example - here you go, a plausible, backwards-compatible "IPv4.1", if you like.

[1] All major desktop OS support IPv6 for last 10 years and major mobile OS for last 7. Let's face it - software upgrade turned out (surprisingly) not to be the biggest obstacle.