For a while now, I've been mulling over my theory about the relationship between Bond and Eve. Now, being an old guy, I mull slowly, and I'm not even sure that the following thoughts are original, though I haven't gone back through all the threads, on this site and others, to check. I've seen a number of questions arise from time to time, and I thought I'd take a crack at them. If this film had been an adaptation of a novel, there would almost certainly have been a more detailed backstory to Bond's relationship with Eve, but the filmmakers wanted instead to plunge us headlong into the action, so we've got to fill in some things for ourselves. For what it's worth, here are some of my thoughts.
Before the start of the film, the operatives of Station T have recovered the hard drive in a way we can only guess at. They know it's valuable and that both MI6 and some opposition force are desperate to recover it. They don't necessarily know what's on it, and there's no reason why they should. The three members of Station T hole up in a safe house in the old city of Istanbul, more or less sealing themselves in while they await instructions from London. M gets the word about the drive's recovery, and she also knows its value. She assigns Bond to go out to Turkey to recover the drive. Eve is assigned as Bond's number two. She's likely to have been drafted in from another station, say somewhere in Eastern Europe, and given instructions to handle Bond's logistical arrangements. She uses her own first name as a contact name, but Bond doesn't know anything more about her. There was no reason for him to do so. His assignment is to land in Istanbul, be picked up by Eve and taken to the safe house, take possession of the drive, and travel back to the airport for same-day return to London.
Eve picks up Bond in the Land Rover and drives him down into the old city. On the way, Bond places a call to Ronson's cell/ mobile phone number to get the all-clear sign. There's no answer on the first call, or on the second, so Bond suspects trouble when he enters the building. In the meantime, Eve has been keeping surveillance from outside, and when she sees an unfamiliar man emerge from a side door, she picks up Bond, and the chase is on.
Questions have been raised about Eve's subsequent suspension from field duty, but I don't regard this as punitive. I think M, particularly given Mallory's interest, has convened an inquiry into the Istanbul fiasco, and Eve, as the only survivor, has been called in from the field to provide her input. She's naturally a bit embarrassed upon meeting Bond again, and he still doesn't have an opportunity to get to know her full name before he's off again to China.
I don't claim that these are great insights, but I think they do add a bit of backstory to the film's opening sequence and are consistent with the action as depicted.