Is it possible to have a romance with all your possibilities in one game?

Solution 1:

Yes, with one major caveat: You can only "consummate" a single romantic interest before the Suicide Mission, though you can dead-end yourself (I.e., prevent further romance) with all of the characters before this.

To observe all romantic interactions within a single playthrough, simply play the game, as normal, until you complete the suicide mission. You can romance one of the available options during this part, but limit your social interactions to only this character (so you don't accidentally lock yourself out of the other relationships prematurely).

After you've beaten the main game, you can go back and start romancing the other love interests. (You may have to do missions / visit planets in between to progress along the dialog trees) Eventually, the new character will refuse to go any further until you break up with your current love interest. If you do so, you should be able to jump directly into the romance scene the next time you speak with them (this is the scene that normally plays just before the Suicide Mission). Add another notch on Sheperd's bedpost, and continue on with the third potential romance.

Do note, you can only be romantically involved with a single character at a time, but you can see all possible romance outcomes in a single playthrough (of a single gender, of course. Fem Shepperd will never be able to romance Tali or Miranda, for instance).

The Mass Effect Wiki has more information on Romance in general.

Solution 2:

No. You will eventually be forced to pick between them. While you can flirt/pursue up to a certain point to actually have a relationship (and the achievement) you must make a choice.

Romance Guide

In addition, when you make that choice the other one becomes quite upset.