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This is an emergency. Do you have an emergency clinic that works with avian species nearby?
She has a prolapse and the bloody tissue is the cloaca and oviduct. There is a possibility of rupture or penetrating injury and subsequent peritonitis. She should be taken for veterinary care immediately. This condition can lead to shock and life-threatening metabolic issues. A vet will need to give her supportive care and try to remove the egg from the prolapse with as little injury as possible, clean and replace the tissues; and start antibiotics, pain control, and other treatments.
You can keep the prolapse moist with sterile saline, artificial tears, gel such as KY. Do not apply anything else or try to pull the egg out yourself.
I wish her the best. Having 2 myself, I am vey fond of Grays.
Do you have a regular vet for her? They might have after hour referral number on their answering machine/ message.