I've had these in my garden for years but only saw the inside of one today. I remember seeing something like this in a salad once.
What is it, and is it edible?
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That is a pomegranate. The fleshy, juicy seed pods are known as arils and are most certainly edible. The outer husk of the fruit and the interior membranes that separate the arils are generally not eaten.
Based on the color, the specimen in your first picture may be a little unripe. The outside will turn red when the pod is fully ripened.
Congratulations! That looks very tasty indeed.