Adventurers often get themselves into political trouble, at least in my games. While the 2e DM asks who would be irritated if the adventurers started hiring knights and other retainers out from under local lords, it doesn't ask who would be irritated if a group of sellswords swooped in and started performing tasks for money. In most fantasy worlds, people can agree on what they want. If a kingdom is beset by a number of ills, there are usually not a number of stances about how to deal with them, or whether or not they are ills at all. That's not the case in the 10th Age which can make everything that much more complicated. It sometimes drives my players to distraction, but I think serves to make a fuller game.
In practical terms, it often means a LOT more work for me. I need to figure out the major factions in each realm, determine who belongs to them, and mark down the most important figures in each. This means that every time a new party starts, I must already have begun to work on the political structure surrounding them. While they usually do not see it at first, it remains suspended there, like an invisible semantic web, and their actions shift and move it. Not every faction thinks the same things are good or right for the kingdom, nor does every member within a faction have the exact same opinion. Kingdoms are not monoliths of power, they are intricate political machines with hundreds or thousands of moving parts.
Can I model all of this? Probably, if I was insane or very dedicated. Do I? Usually not. I try to model the top level of the machine first of all: the high nobles, the great councillors, the major temples. It determine what they want collectively, what kinds of philosophies of governance they hold, and who they like or dislike. That serves as a good background for me to delve into an individual region. Now, local barons and magistrates usually do not form factions of their own (they may) but usually attach to the faction of a larger party. This means I can say that this here little baron and that one are attached to this or that major faction. It's very easy in lands were blood-relations make alliances, since all I have to know then is what house they owe their bloodline to.
Now, I usually also add mitigating factors for smaller players. Such as, "Supports the Aderno family but is not active in politics," or "Supports the Emperor nominally, but believes that the court is highly corrupt." This allows them wiggle-room to be convinced, cajoled, manipulated, or allied with against their own supposed interests... if the PCs are clever enough to discover exactly how to do it.
The effect this has on the game is one that builds up over time. While almost no one could object to adventurers clearing out, say, a basement full of necromantic horrors in the capital, they might begin to suspect that the adventurers are acting on the orders of some faction they don't like. Or perhaps they begin to think that the adventurers are forming their own political faction, which is just as bad. Or mayhaps they don't care, but the adventurers actions are having an effect on their own plans—for example, defending the status quo almost always upsets the people who hate the king (see Someone Always Hates the King).
There are no politically neutral actions in a machine of this complexity. Everything has effects, even if they are not immediately seen. The effects lead to fame, glory, alliances, but also foes, assassination attempts, and permanent enemies. Depending on what exactly the party does, these may come earlier or later in their level-arc. It's possible for a 1st or 2nd level party to aggravate a high noble to the point where he dispatches murderswords to come and kill them in their sleep. Likewise, it's possible for that same party not to aggravate anyone for 6-9 levels. Or perhaps to become so famous in their 1-2nd levels that they begin to attract followers and hirelings of their own accord early.
The important part isn't how exactly they interact with the machine, it's that this level of play exists at all. This invisible, sticky web of plots and politics may be unseen, but to remove it entirely is to remove a whole level of complexity from the game. And you now I'm almost always for more complexity, particularly when it doesn't involve math.