RSVP: Droman Aliothe

Updated as of Patch 9.0.5 (using the steps from Tig3rE's post):

Take the Quest RSVP: Droman Aliothe from Droman Aliothe in the Ardenweald in Hibernal Hollow. The big bear near the flight master.
/way 60.11 53.72
To complete the quest you will need [item=180774], which is sold by Ta'ruca in Revendreth at the Night Market.

1) Go to the Night Market in Revendreth to Ta'tru, southwest of Darkhaven.
/way 51.11 78.81
Buy a [item=180973]
- the price changes every week
- e.g. 20x Laestrite Ore or Shadowy Shank
- you can farm it yourself or buy it from the AH

2) After that, fly to Oribos and go to the The Broker's Den, which is in the basement of "The Idyllia". It is the eastern entrance in the "Ring of Fates", the one with the beer mug above the entrance. Then go into the right or left room and through the door down the stairs.
/way 51.20 44.04
Trade the [item=180973] for a [item=180971] at Ta'lan the Antiquary.

3) Then go back up to the "Ring of Fates" and turn left into the Hall of Curiosities.
/way 61.84 71.82
Trade the [item=180971] for a [item=180970] at Acquirer Ta'gosh.

4) After that fly back to the Night Market in Revendreth.
/way 51.11 78.56
Trade the [item=180970] for [item=180774] at Ta'ruca.

Finally fly back to Droman Aliothe in the Ardenweald and complete the quest to get RSVP: Droman Aliothe.

As of 9.0.5 this quest removed the Bastion and Maldraxxus steps. The updated trade-chain looks like:

Buy Necklace of Dredbat Fangs from Ta'tru for various crafting materials
Go to Oribos and trade Necklace of Dredbat Fangs for Aquamarine Cartel Chit from Ta'lan the Antiquary in the downstairs Cartel room
Use Aquamarine Cartel Chit to buy Teregeer Crystal from Acquirer Ta'gosh in Oribos' Southeast wing
Take Teregeer Crystal back to Revendreth to buy Preserved Berries from Ta'ruca
Take Preserved Berries to Droman Aliothe for the RSVP
(Note: quest need not be accepted to begin the trade chain, you can do the entire chain without visiting Ardenweald prior to the turn-in)

I highly suggest that when you do the quest for the first time to get several of the Preserved Berries. You need to redo this quest every time you want to invite Droman to your party.