I have managed in the end to make omnicompletion to work. I have added g:gitlab_api_keys variable:

let g:fugitive_gitlab_domains = [
    \ 'https://gitlab.com',
    \ 'https://gitlab.internal.site.com'
    \ ]

let g:gitlab_api_keys = {'gitlab.com': 'somethingverysecret'}

Then I had to discover that appropriate remote for git must be origin (that's a pity, I usually use upstream as a name of my remote; but I can survive).