Here is the solution to empty the supply on turn 1 of player 5 in a 5-player game.
(I left out the last part, which is very similar to CC's solution to empty the supply)
Kingdom: Lost City (LC), Messenger (Me), Stonemason (SM), Masquerade (Ma), Procession (Pr), Fortress (Fo), Catacombs (Ca), Haggler (Ha), King's Court (KC), Watchtower (Wt), Black Market (BM)
Black Market deck: Baker, Young Witch, Prince, Coppersmith, Graverobber, Rogue, ...
Events: Burrow, Travelling Fair, Mission, Ball
Shelter game
Starting hands:
P1 draws 5C
P2 draws 5C
P3 draws 5C
P4 draws 2C,N,H,O
P5 draws 2C,N,H,O
----- Player 1: -----
P1 plays 5C, spends baker token, buys Burrow, buys SM1, gains LC1 and LC2. Everyone else draws 2C.
Cleanup: P1 draws C,C,O,N.
----- Player 2: -----
P2 plays 7C, buys Me1. Everyone gains Ma. (P2 gains Ma1, P3 gains Ma2, etc.)
Cleanup: P2 draws H,O,N (reshuffle),Ma1,Me1.
----- Player 3: -----
P3 plays 7C, spends baker token, buys Burrow, buys Travelling Fair, Mission, Ma6.
Cleanup: P3 draws N,O,H (reshuffle), Ma2.
----- Player 3 (mission turn): -----
P3 plays N, plays Ma2, draws Ma6, C, passes: (1 C, 2 Ma1, 3 H, 4 H, 5 H), trashes O, draws C.
P3 plays Ma6, draws 2C, passes: (1 C, 2 O, 3 Ma1, 4 H, 5 H), trashes O, draws C.
P3 plays 5C, buys Ball, gains Pr1, Pr2.
Cleanup: P3 draws 2C (reshuffle), Pr1, Pr2, Ma2.
----- Player 4: -----
P4 plays N, plays Ma1, draws 2C, passes: (1 O, 2 Me1, 3 Ma2, 4 C, 5 H), trashes O, draws C.
P4 plays Ma2, draws (reshuffle) Ma3 (deck empty), passes: (1 N, 2 O, 3 Pr1, 4 Ma3, 5 N).
P4 plays 6C, buys Mission.
Cleanup: P4 draws (reshuffle) N,Ma1,Ma2,Pr1,C.
----- Player 4 (mission turn): -----
P4 plays N, plays Ma1, draws 2C, passes: (1 N, 2 N, 3 Pr2, 4 Pr1, 5 O).
P4 plays Ma2, draws 2C, passes: (1 O, 2 N, 3 Me1, 4 Pr2, 5 C).
P4 plays 5C, buys Ball, gains Fo1, Wt1
Cleanup: P4 draws C (reshuffle), Fo1, Wt1, Ma1, Ma2
----- Player 5: -----
P5 plays Pr1-Pr2,
- plays Ma3 twice, draws 2C, passes: (1 C, 2 N, 3 N, 4 Fo1, 5 C), draws C (reshuffle), Ma4, passes: (1 C, 2 C, 3 N, 4 Wt1, 5 C), trashes Ma3, gains Pr3.
- plays Fo1 twice, draws (reshuffle) Pr3 (deck empty), trashes Fo1, puts Fo1 in hand, gains Ca1, reveals Wt1, trashes Ca1, gain Pr4.
- trashes Pr2, gains and trashes Ca2, gains Pr5
P5 plays Ma4, draws (reshuffle) Pr4, Pr5, passes: (1 C, 2 C, 3 C, 4 C, 5 C)
* Now P5 can start its big Procession chain, playing Procession whenever possible, and Fortress otherwise. The first gain priority is Procession (via Catacombs), and when the Processions are out, start gaining Hagglers and a single Black Market (via Catacombs) and possibly another Watchtower.
* Then P5 can play most Hagglers, playing Wt for draw. Then play a BM, buying a card costing at least 6, gaining the remaining Hagglers, and as much Watchtowers and Black Markets as needed. If P5 needs more draw or actions, P5 can gain some Lost Cities.
* Then P5 gains all KCs by buying a single Prince from the BM deck. Now KC the remaining BMs to gain as much cards as possible, gaining a huge number of cards, including Coppers. Buy a Scrying Pool and Counting House for draw. Buy a single Coppersmith and KC it for coins.
* For maximal coins, P5 must keep 2 in the supply during the Procession chain, and Process Coppersmith twice, getting it back with Rogue and Graverobber. Also, you want to play some Masquerades to get some Coppers from the other players (and give some junk back, e.g. Stonemasons gained with Haggler).