Alpha Protocol

The following solution features SPOILERS...

There are four possible romances in the campaign and as per the description of the achievement they must be completed in one playthrough (each individual romance yields its own achievement as well as in-game perk based bonus to Mike's endurance).

The ladies of interest are:

Mina - your handler and Alpha Protocol analyst
Scarlet - freelance photojournalist (with a secret secondary occupation)
SIE - VCI mercenary (and a cougar with a big gun)
Madison - VCI employee (with a secret family connection)

Due to the complexity of the game, the multitude of the objectives and non linear plot progression it is rather difficult, without multiple playthroughs, to pinpoint all romance triggers and requirements. As such this solution provides general pointers towards the accomplishment in addition to the notes on how my own Mike managed to find his way into the hearts of those lovely ladies.

The game tracks Mike's relationship with NPCs (including the ladies) via reputation points (positive or negative) that range from -10 to +10 (starts at 0). I can only assume that each romance has a minimum requirement in the reputation to be triggered and while the exact number is unknown to me I will provide you with a number and a state of the relationship that was sufficient for me to obtain the achievement. Should you be able to narrow the details further please feel free to post your discoveries in comments so that the solution can be updated.

Mina

Location - Alpha Protocol HQ , Primary Handler for majority of missions
Trigger event - after completion of 4 operations (including Saudi Arabia) Mike relocates to the States to carry out Operation Endgame against the Alpha Protocol, Mina will greet him in person at the Alpha Protocol safe house upon his arrival at which point opportunity will present itself to kiss the lady to start the office romance...
Reputation - romance achieved successfully with reputation of +9
Preferences - Mina prefers Professional (red) approach though she is open to occasional suave/flirting (blue) approach, assuming it is tasteful. Mina can be a little bit jealous at times...
Special Notes - be nice to your handler and follow her suggestions/advice and you should do fine
Personal Note - Mina is my favourite lady in the game, my version of Mike “rode into the sunset” with her at his side (after cleaning up all the mess of course)

Scarlet

Location - Taiwan Hub only
Trigger event - after completion of 4 operations (including Saudi Arabia) Mike relocates to the States to carry out Operation Endgame against the Alpha Protocol, Scarlet romance specific mission (visit to the hotel) should become available at this time, Scarlet may ask Mike to stay for “behind the closed door interview”...
Reputation - romance achieved successfully with reputation of +10
Preferences - Scarlet prefers professional (red) and suave/flirting (blue) approach, though it is recommended to keep things professional when she accompanies Mike on the mission, Scarlet is a thrill seeker, try not to bore her with your replies and you should do fine
Special Notes - during my playthrough I have forwarded all mission obtained intelligence (in form of e-mails) to Scarlet for publishing, this may or may not have had an effect on the outcome of the romance, though it is worth mentioning that Scarlet suggested we should meet for a dinner at some point in one of her replies.
Personal Note - remember to discover more about her secret secondary occupation during the Operation Endgame (for extra achievement)

SIE

Location - Russia Hub (primary), other location (based on your choices)
Trigger - during the Operation Endgame Mike surrenders to Alpha Protocol as part of his infiltration plan, at one stage he will find himself strapped to the table when SIE shows up... she is certainly willing to free Mike, but not before she satisfies her needs...
Reputation - romance achieved successfully with reputation of +10 (possible with +7 as per third party comments)
Preferences - SIE likes aggressive (yellow) approach, even if it seems ridiculous at times she will always interpret it the way she wants to, however you are advised to walk carefully when engaging SIE and others in the same mission/conversation
Special Notes - during my playthrough I have favoured SIE (VCI) over Albatross (G22) when opportunity arose, this does not mean you are forced to become hostile towards Albatross/Sis... the trick is to walk the fine line. In Italy I refused to work for Conrad (SIE hates Conrad). In addition during Moscow end missions I chose to rescue SIE from her predicament before proceeding with the objectives. Finally I have selected SIE as the initial handler for the Operation Endgame (according to 3rd party comments this is necessary).
Personal Notes - rawr dahrling!

Madison

Location - Italy Hub only
Trigger Event - Once Mike meets Madison in person he should offer her the protection at which point Madison will re-locate to the Safe House, after few missions Mike will arrive late at night and find Madison asleep, if he chooses to check up on her the conversation may lead to much more “checking up” that it was initially intended...
Reputation - romance achieved successfully with reputation of +4 (Madison does not play a very large part in the game and interactions with her are limited)
Preferences - a mixture, mostly professional (red) and suave/flirting (blue)... depending on the situation and conversation
Special Notes - during my playthrough I have selected Madison as my handler for the mission in the Conrad’s Villa (this might not be necessary)
Personal Notes - the difficult decision here lies in the mission AFTER the romance have been completed... keep her family connection (once you find out about them) in mind when making your choice


Hope this helps... if you have any specific questions please feel free to PM me at your discretion...

Personally I found mina and madison required little thought, as they just kind of happened anyway.

With SIE, on my first run I didn't help her at Konstantine Braykos mansion, when given the option by mina to go for the Data, or for SIE, make sure to pick SIE. (If you selected her as handler for the mission)

With Scarlet, when you meet her in the taiwanese hotel, make sure to be 'honest' with her throughout, otherwise she won't offer for you to stay