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Could a land mass reasonably exist where I want it on my earth-like binary planet?

Original question on worldbuilding.stackexchange by Ray Lawler

I’ve been doing a lot of research lately on my binary planet system for the series I’m writing. Some details I’ve already worked out:

The system’s star is 0.6 solar masses, 0.8 solar radii and 0.3 solar luminosities, making it a K-dwarf.
My planetary system is orbiting 0.5 AU from the star with nearly no eccentricity.
Both planets are earth-like in size and ability to support life. They are tidally locked, each with one side always facing the other.
The planets are roughly 10x closer to each other than the moon is to earth. With that and being earth sized, the planet appears about 37x larger in the sky than our moon does at the nearest point. (This is outside the Roche limit.)
Because the planets are tidally locked to each other, tides are solely caused by the sun. Due to the size and proximity of the K-dwarf star, tides on each planet are approximately 1.5 times the magnitude of the tides on earth (from the sun and moon).
The incredible constant pull each planet has on each other results in massive bulges centered on the equator at the points closest to and furthest away from the other planet. This bulge (assuming I’ve done my math right) has a height equivalent of a tide 56,462.43x stronger than a tide on earth, which at its high points would be 33,808 meters deep, or nearly four Mt. Everests deep.

I have other information that I don’t think is relevant to my questions. If someone thinks other information is needed to answer or provide as useful context, let me know.

My main question is this:

Could a land mass realistically exist above water at a point on one planet that is closest to the other planet (where the water would typically be deepest)?

No matter what I do for my books, I need there to be people living around this area, so if I have to make it a floating island of some kind I will. I’ve considered a massive breakoff from a bayou or a modified pumice floating island, but those don’t really satisfy my desire for a real, massive, island that actually connects to the ocean floor and has its own volcanoes. So, would it be reasonable for land to form basically 4 Everests high, enough to break out of water at the highest point in its bulge? If not, I’ll need to come up with another way to have a land mass there.

Answer by Benjamin Hollon

Let’s take a look at this.

As AlexP points out, tidal forces work on both the water and the crust. So we don’t really need to worry about it being a huge amount deeper on that side.

But if that’s not enough, we have a secret weapon...

Volcanoes

Volcanic islands should be able to make you some nice islands even in places where the water is deeper. In fact, I’d expect there to be more volcanic action than, say, on Earth because the tidal forces are messing with the crust (though I’m not a volcanologist, so take that with a grain of salt).

Just imagine: Hawaii (volcanic islands) with a beautiful view of a planet taking up 37x the amount of sky the moon does. I’d say the tourism industry would be pretty strong to a place like that. I know I’d pay for it. ;)