I spent most of SHL3 trying to get a competitive Foundry + Twins deck working. I'm throwing my findings - such as they are - open for discussion in the hope of "refining the process" (ahem). I enjoyed playing all the variations of the deck enormously and pulled out some good results but, even allowing for my rather erratic level of often sleep-deprived/screaming baby distracted play skill, I still don't think Foundry is quite "there".
I played 34 games in the league with variants of the deck below, plus casual games on top. My win rate started very high - when Order and Chaos was very new and runners were still generally the underdogs - but it fell as runners gained confidence and as I made some unsuccessful changes. It bottomed out at 55%.
Games generally went one of four ways:
1) Win. Runner faceplants into Merlin plus the Twins for 8 net damage in the first few turns of the game.
2) Win. I manage to get an Eve or Adonis to stick and either score out behind a combination of ASH and/or The Twins or the runner is pressurised into running a Twins server that wrecks their rig/economy or flat-lines them.
3) Loss. The runner gets a strong economy start and trashes my economy assets on sight. I never get the economy to secure my servers.
4) Loss. Set up a secure remote server and about to score out, but lose to multi-access - usually on R&D (and often because of dubious ICE placement on my part).
In terms of match-ups, the deck had a very strong record against PVP Kate (with something like an 8-2 win rate). The ability to lock Lady out of a server using multiple Galahads (especially in combination with The Twins) combined with being able to spam remote servers guarded by code gates allowed the deck to put a lot of pressure on Kate's recursion, opening up scoring/flat-line windows. It also had a strong record against Reg-Arse MaxX (4-1 I think), but I attribute that primarily to people being new to the runner (and the brief inclusion of Cyberdex Virus Suite). Its worse match-up was Eater MaxX - with Eater and Wanton Destruction both being terrible news for a Grail deck. It never won that match-up.
The Deck Itself
So, the deck itself in its most successful form:
The Foundry: Refining the Process (The Spaces Between)
3x Accelerated Beta Test (Core Set)
3x Priority Requisition (Core Set)
1x Project Vitruvius (Cyber Exodus)
1x Utopia Fragment (The Source)
3x Adonis Campaign (Core Set)
3x Eve Campaign (Humanity's Shadow)
3x Jackson Howard (Opening Moves) •••
3x Ash 2X3ZB9CY (What Lies Ahead)
3x The Twins (Order and Chaos) •••
3x Hedge Fund (Core Set)
3x Restructure (Second Thoughts)
3x Eli 1.0 (Future Proof)
3x Galahad (Upstalk) •••
Code Gate (8)
3x Enigma (Core Set)
2x IQ (First Contact)
3x Merlin (All That Remains) •••
3x Ichi 1.0 (Core Set) (But see below)
3x Lancelot (First Contact) •••
15 influence spent (maximum 15)
20 agenda points (between 20 and 21)
49 cards (min 45)
Cards up to Order and Chaos
This is already very long, so I won't dwell on individual choices too much.
At @SimonMoon's suggestion I tried Corporate Troubleshooter instead of ASH. Although it scared the c**p out of people, I found having to spend the money earlier in the run meant firing it was very expensive relative to ASH. It also doesn't in itself end the run which was often problematic with shapers and anarchs losing their breakers to lancelot, only to clone chip the relevant one back out and score. I would possibly include it as a one-off in a fixed-breaker heavy environment, but no more (to be revisited when NEXT Gold comes out). I also tried IT Department, but was unconvinced - the Foundry's ever weakening R&D made spending turns pumping power counters feel unattractive.
The second sentry slot could be one of Ichi 1.0, Ichi 2.0, Architect or Janus 1.0. With Mimic everywhere Architect feels a little expensive in Foundry, where you don't get the credit for installing on the opponent's turn. Ichi 2.0 is stronger against Atman but weaker against D4V1D. Janus has a lot going for it - a second flat-line route, a massive tax if rez'd, increasing the threat of ABT/Pri Req. But getting to 17 credits is a long slog and having it in the deck exacerbates the economic weakness that is the biggest thing holding the deck back. On balance, Ichi's reasonably priced 3 credit tax, plus moderate Twins threat, is probably the most consistent choice.
Otherwise I think the choices are broadly self-selecting. NEXT Ice is too vulnerable to parasite for my liking, although I will revisit that when Gold comes out - Enigma and IQ are both decent replacements.
You are looking to get an economy asset ticking over, or score when the runner is broke from trashing your economy. This isn't ETF, you can't get away with rezzing naked Eves and expect them to stick so expect to be rezzing stuff to keep them around. Think very hard about your early ICE placement - this will have profound impacts on the game later. A Twins plus Merlin server should go down as early as possible - you'll be amazed how many people will kill themselves on it - but hold the other Merlins if you can. Then build a scoring remote whilst defending the "problem" central (R&D for Shaper, HQ for Criminal, for Anarch it depends on the build). Be aggressive, especially whilst they are rig building. Your late game is poor because of Foundry's drain on your R&D, but your mid-game is brutal if set up properly.
Two variants that I didn't get a chance to fully try:
@Calimsha suggested a more rush-based build, with a more operation focused game and a 2/2/2/1 agenda scheme. This is something I'm primarily looking into trying when NEXT Gold comes out. Prior to that, this is the build I came up with.
I also tried a Blue Sun build, using the synergy between Blue Sun and Grail ICE to remove one of the decks bigger weaknesses - having to play ICE in sub-optimal positions when you would rather keep them in hand. Notably mostly for actually winning games due to Levy University, but I was not convinced it had the consistency to be viable.