Gil is a character that you won't be able to take to the battlefield - he is one of the Tempest crewmates. However, he, as a romance option, is only available to the male character, Scott Ryder - while playing as Sara you won't be able to flirt with him and the relationship will be limited to friendship. Gil Brodie Anwar is a character you can have close relations with - by engaging in that relationship you won't be able to flirt and romance with other characters. You can flirt with him each time you finish a main mission (Priority Ops).
Note - before engaging in a close relationship you can save the game. Afterwards, you can go through the rest of the romance, load the saved game and engage into a relationship with a completely different character. The option to romance other characters is locked only when you are engaged in a close relationship with somebody.
Below you can find a table presenting all of Gil's loyalty missions, coupled with the requirements that must be met to unlock them. Detailed description of every single one of them can be found in "Allies and relationships" section of this guide.
How to unlock
Gil Brodie: A Game of Poker
Finish "A Trail of Hope" mission.
Gil Brodie: The Friend
Finish "Hunting the Archon" mission.
Note: the option to engage in a close relationship will appear during "Gil Brodie: The Friend Family" loyalty mission. Remember that this choice will prevent you from flirting and romancing other characters. If you still want to romance with others and at the same time see the final parts the romance with Gil, save the game before starting "Gil Brodie: The Friend Family" mission and load it later if needed.