I've handled the major relationships typically between sessions.
Most sessions are committed to adventuring time, the between session emails and talks over coffee are for background character development.
That said, the best relationship (in a group with 5 PCs), was when the group's wizard saved a woman (M'tilde from Hot Day in L'Trel (sp?) modified for Longspear in Greyhawk). He kept track of her and her baby through the long month and helped get M'tilde more active as a character (given her prior military background). Eventually she agreed to be a henchman/cohort when the group moved on to Greyhawk City. After several adventures on the road, and the constant attention of the wizard (no romance on-screen), the player thought it would be good to have them marry. We set up some things in our down-time, and in a few game-months (just a few sessions in real time), we had them marry. His continued help (and various plot hooks) end up with her establishing the guild in Greyhawk for tailors as one of the more significant guilds. She was an outlet for the PCs desire for politics (as a major guild leader, she was on the council), and the players had a hook into politics, but got to keep their preferred focus in adventuring (dungeon dives and wars).
Generally she helps out the wizard as often as he helps her out. It takes some restraint to not have her be a strict liability for the party (of course, fortifying their home and hiring guards for M'Tilde helps), but was generally fun.
Again, not much game-time romance, the between-game discussions were more of the "what does she want/need/etc" variety - kept us from role palying a m/f relationship when we were both males (and likely both would have been a little uncomfortable).