A newly discovered moon tunnel

When such discoveries are made who "owns" that real estate?

Whomever can occupy it. Outer Space Treaty prohibits national claims but not use for peaceful purposes. So if you sit on it, you own it.

TLDR:

Space nerds theorized moon caves since the 80s. Moon caves created by young moon lava flows billions of years ago found in early 2000s. Japanese space agency finished mapping a tunnel last week and found it's 50km long and 500m wide with a flat bottom perfect for building space buildings in.

This is news that's breaking to a revitalized moon rocket industry with all agencies and players building their own big moon rocket. New Glenn for blue origins, bfr for spacex, sls for NASA, ACES for ULA (boeing aerospace + lockmart ) Chinese have also expressed interest in a moon base.

For a more technical description, the paper is here: http://onlinelibrary.wiley.com/doi/10.1002/2017GL074998/full.

"Detection of intact lava tubes at Marius Hills on the Moon by SELENE (Kaguya) Lunar Radar Sounder"

Intact lunar lava tubes offer a pristine environment to conduct scientific examination of the Moon's composition and potentially serve as secure shelters for humans and instruments. We investigated the SELENE Lunar Radar Sounder (LRS) data at locations close to the Marius Hills Hole (MHH), a skylight potentially leading to an intact lava tube, and found a distinctive echo pattern exhibiting a precipitous decrease in echo power, subsequently followed by a large second echo peak that may be evidence for the existence of a lava tube. The search area was further expanded to 13.00–15.005°N, 301.85–304.01°E around the MHH and similar LRS echo patterns were observed at several locations. Most of the locations are in regions of underground mass deficit suggested by GRAIL gravity data analysis. Some of the observed echo patterns are along rille A, where the MHH was discovered, or on the southwest underground extension of the rille.

I'm only familiar with Bird Law, and as there are no birds on the moon, I am sure that "rock law" will predominate in that if I hit you over the head with a big enough rock, I then obviously own the moon.