Probability that one part of a randomly cut equilateral triangle covers the other without flipping

At Probability that one part of a randomly cut equilateral triangle covers the other, the case with flipping allowed was quickly solved. The case without flipping seems more difficult and hasn’t been adressed, so I’m posting it as a separate question:

What is the probability that randomly cutting an equilateral triangle will allow one part to cover the other if you’re not allowed to flip the parts?

The cuts are distributed according to Jaynes’ solution to the Bertrand “paradox”: random straws thrown from afar, with uniformly distributed directions and uniformly distributed coordinates perpendicular to their direction.

A succinct characterisation of the cuts that allow one part to cover the other would already constitute significant progress.

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