Note: Much of the information here is redundant, as this quest was functionally covered during Passage to Anywhere. Still, it'll get its own page like any other quest for you silly readers who aren't following the walkthrough chronologically... well, as chronologically as possible, anyways...
After you get the Unreliable up and running the quest Passage to Anywhere will start, which is frankly optional. It's wise to do it, and starting it properly means you'll get a chance to start this quest.
Fly the Unreliable to Groundbreaker and talk to Cpl. Leonard Wheeler to find out that your ship has been impounded by the Byzantium representative, Udom Bedford. Not a surprising development if you read through Hawthorne's terminal on your ship.
Pay a visit to Udom Bedford in the Byzantium Embassy, which is along the northeastern end of the Promenade on the Groundbreaker. Getting him to release your ship from impound is a trivial matter, as he clearly imposed it just to force you - or rather Alex Hawthorne - to speak with him. Afterwards he'll bring up Phineas Welles, and this quest will begin if you agree to turn in Phineas.
There's only one small problem... Udom is an unscrupulous idiot, and sold his official seal to Gladys, a black market dealer on Groundbreaker. Without that seal he can't send you to the proper authorities back on Byzantium. You know, people in a position to actually act on whatever information you care to divulge.
That being the case you'll have to talk to this Gladys, who will sell you the seal for 8,000 Bits. It's a hefty sum of cash, but don't worry - she's got a way for you to make money. A job offer, in other words. She intercepted a radio transmission from Roseway, an Auntie Cleo site on Terra 2 that was supposedly abandoned years ago. Whomever is there might just have something to hide, and secrets tend to have buyers. This starts the quest The Distress Signal, which is a side quest that leads you to a new area to explore, with quests to complete, enemies to kill and most importantly for the task at hand, Bits to earn.
Whether you go to Roseway or not, you'll need to raise the 8,000 Bits somehow. Once you do, buy back the Official Seal of Halcyon Holding Board from Gladys and return it to Udom Bedford to complete this quest and start The Puppet Masters. This not only nets you the Byzantium Navkey, which you'll need to travel to Byzantium on Terra 2, but some Byzantium Travel Authorization Papers, which entitle you to land at Adjutant Akande's Landing Pad.
Note: Completing this quest does not conflict with Passage to Anywhere or The Distress Signal. You can get Udom's seal back, travel to Byzantium and do whatever you wish there, then go to Roseway, do whatever you care to do there, buy the Stellar Bay Navkey from Gladys and continue working with Phineas if you so please.