You can retrieve Ethereum private key of your Eidoo wallet using the backup phrase (12 words) through an external tool. Follow the directions below:
1. Connect to https://iancoleman.io/bip39/ (for greater security the tool also works with the computer off-line)
2. In the Mnemonic section type the 12 words in the BIP39 Mnemonic field and set the Coin field to ETH-Ethereum
3. In the Derivation Path section choose the BIP32 tab and set the BIP32 Derivation Path field with this value:
4. Check that all data you entered is correct and moved down in the Derived Addresses section
5. Your private key is the one you find in the first line of the table, under the Private Key column.