If there's still anything left in orbit (e.g. dead satellite), an optical or radio signal could be passively bounced off it, e.g. https://ham.stackexchange.com/questions/8897/could-uhf-earth-satellite-earth-bouncing-of-signals-off-of-this-large-metallized ([EDIT] For this, you would want your bounce satellite to be in a https://en.wikipedia.org/wiki/Graveyard_orbit so that it's around for a good, long time.) Similarly, it's also possible to bounce signals off the moon, if that's still present, but the increased distance makes it more challenging. This is probably the highest bandwidth option.
Between ocean bases, something similar to the ELF radio system used for submarine communications may be usable , see https://en.wikipedia.org/wiki/Communication_with_submarines#Extremely_low_frequency. However, only a few letters per minute can be sent (not a typo) so it is very limited.
Depending how sci-fi you're willing to go, communication using neutrinos has been demonstrated to be possible: https://physicsworld.com/a/neutrino-based-communication-is-a-first/. Because neutrinos pass through matter easily, they can go through the entire planet. However, that also makes them extremely difficult to detect, making the both transmitting and receiving equipment extremely bulky and complex. The data rate would be rather low (maybe a handful of kilobits per second?) but you could probably fudge a slightly higher data rate without the reader noticing.
[EDIT] Without a terrestrial fiber or tower network or an active satellite network, I can't think of any way that anything like today's Internet speeds would be practical in the proposed scenario. In the cases listed above, I'd expect communications something like the early 1980s Internet, where messages are completely text.