The origins of BhadraKaali + Is she the consort of Veerbhadra?

During my journey of studying Shaiva and Shakta scriptures I have found mention of instances where Veerbhadra's implied consort is BhadraKaali as they both arose from Lord Shiva's fury when Mata Sati sacrificed her life in the sacred havan fire during Daksha Prajapati's yagna for Lord Vishnu. Veerbhadra and Bhadrakaali were instrumental in teaching Daksha a lesson for insulting Lord Shiva.

There is also a story - it escapes me where I heard it but BhadraKaali is said to have had an alternative origin and was not the consort of Veerbhadra.

Can someone help me out in understanding this topic better?

Har Har Mahadev!