Can an Unchained Monk use Flurry of Blows while in control of a grapple?
I think one can, due to this FAQ, but I'm not certain, as the source link goes nowhere.
This Grappling Flowchart implies that a controlling grappler can only take a standard.
So which is it? A standard, or is a full attack possible?
Maintaining a grapple is normally a Standard Action. The FAQ you linked does not contradict this, but he is talking about a monk character being grappled, which normally limits your actions since you have one hand unavailable, and since it doesn't matter for monks, they can still make a Flurry of Blows while grappled, but not if they are maintaining a grapple.
Grappled is simply a condition that is applied when characters are in a grapple, and it does not limit what type of actions you can do, other than those listed in the condition:
A grappled creature is restrained by a creature, trap, or effect. Grappled creatures cannot move and take a –4 penalty to Dexterity. A grappled creature takes a –2 penalty on all attack rolls and combat maneuver checks, except those made to grapple or escape a grapple. In addition, grappled creatures can take no action that requires two hands to perform. A grappled character who attempts to cast a spell or use a spell-like ability must make a concentration check (DC 10 + grappler’s CMB + spell level), or lose the spell. Grappled creatures cannot make attacks of opportunity.
The action required to maintain a grapple is a Standard Action, and the only upgrade available to players is the Greater Grapple feat, which converts it to a Move Action instead.
Like if you happen to have the grab special ability (which allows you to initiate a grapple on your target as a free action when you make an attack), like a serpentfolk/naga monk, you no longer require to spend a standard (or move) action to grapple your target again and can freely make full attacks or flurry of blows against her. Another possibility is using Snake Style in combination with Hamatula Strike to grapple opponents hit by your piercing unarmed strikes.
Do note, however, that this does not allow you to maintain the grapple as a free/swift action. That ability is unique to certain creatures (usually ones with lots of tentacles), like the Khala or Atuikakura. But your target will have a very hard time trying to get away from you, as the only moment it will be free is between your attacks. So the only option they have to escape is that you miss your grapple checks or they ready an action to move away whenever you release the grapple.
