The campaign is set in an open world, giving the players lots of freedom, including the option to split up.
This specific player was unfortunate enough to become plagued with a terrible illness, and to prevent the rest of us from getting infected, he decided to leave us and look for a cure himself. Ever since this has happened, the sessions have become awfully slow.
Because we split up, we have to take turns in playing, which isn't an issue on its own, and has always went smooth. However, whenever the player with the plagued character takes his turn, he spends a tremendous amount of time role-playing, and sometimes refuses to take action to make his part of the story progress, preventing all of us from playing. His role-play is over-dramatic and extremely slow. (Specific examples: Coughing every third word, pausing before starting a new sentence, often out-right refusing to take action out of IC stubbornness, etc.)
Furthermore, he often tries to RP things as epic as possible, seemingly like he's playing in a movie. He very often makes stupid decisions based on "How cool or dramatic that could be!" or how likely that could happen in another story, sometimes even mentioning that specific story.
The DM acknowledges this issue and tries to encourage him to take action when he's idling, but he doesn't seem to understand the hints the hints the DM gives off, and ignores the hints the NPCs are giving him (Which is another problem altogether). He wouldn't mind talking with the player about this, but doesn't know how exactly to formulate this issue.
An example of slow IC decision-making which actually happened:
DM: "The ground shakes. Everyone, roll a Reflex Save."
Everyone makes it.
DM: "You see a hole in the ground, with an underground river running underneath your feet. It seems to be heading into the direction of your destination. What do you do?"
The players proceed to discuss jumping in the hole. The sentence repeated most often being: "It would be so stupid to do this... But wouldn't it be awesome!?"
The DM tries to encourage the players into making a decision by having one NPC known for her recklessness and deceptive stupidity (à la Joker) jump in and every other NPC telling them what a ridiculous idea jumping in would be.
After twenty minutes which felt like ages for the players not involved, they finally decide to jump in because that would be awesome.
In this specific example more people were involved, including the problem-player. The same session in which this happened, one specific player who split off of the group and was supposed to make an entrance later, never got a turn, because the session had to end before she could.
The DM doesn't like limiting his players, so he rather not use time-limits. This has been house-ruled.
Private-sessions are an option available to all players. They simply don't seem interested or take them as seriously as the "real" sessions.