COD goes without hesitation to 20dn for its utterly brilliant use of “spaff” as an anagram indicator; but construction of the day to 1dn’s “with secondary deductions”, an amazing spot for a seemingly hard-to-clue word like SCHIZOTYPY indeed. My hat, as ever, off to the setter!
ACROSS
1 Destroyer in a lake overlooked by spit (4)
SIVA – take S{al}IVA [spit], and have it overlook A L. Siva the Hindu destroyer deity, part of a trinity with Brahma the creator and Vishnu the preserver.
3 Scot’s variation in complexion if recently perplexed (10)
FERNYTICLE – (IF RECENTLY*). This is just a Scots word for a “freckle”. Does that count as an “obscure foreign word clued by an anagram”? 
keriothe would know. I will say that deciding where to put the non-crossing letters without looking things up was particularly difficult in this case and, reader, I looked things up.
9 Partially privatise hospital conception system, getting around contrary opponent (4,3)
HIVE OFF – H + IVF [in vitro fertilisation = conception system] around reversed FOE. First one in, as IVF came to mind quickly.
11 The creature I spotted in greasy spoon was a louse (7)
CAITIFF – IT I [the creature | I] in CAFF. Second one in, as it had to be CAFE or similar.
12 Faith in Spanish retailer of clothing, so svelte with small inset (13)
ZARATHUSTRISM – ZARA THUS TRIM, with S “inset”
14 Platter with food originally banned in strict diet (5)
TRAYF – TRAY + F{ood}. The opposite, I believe, of kosher.
15 One rarely takes action against head demon? (9)
IMPLEADER – or IMP LEADER. As suggested, a rare word for one taking (legal) action.
17 Strait-laced person adopts uncertain expression to hold over French conciliator (9)
PRUDHOMME – PRUDE “adopts” HMM “holding” O. I had assumed this was a historical Prudhomme such as Louis-Marie or Sully, not stopping to worry about what exactly they might have conciliated. But I was wrong! A prud’homme is, in France, “a member of a board for settling labour disputes”.
19 Classically elegant trellis with sides lopped (5)
ATTIC – {l}ATTIC{e}
21 Weber composed Erika with discordant framing in unspeakable style? (13)
JAWBREAKINGLY – WB + (ERIKA*) “framed” by JANGLY. I think a jawbreaker here is just a word hard to pronounce, rather than something that renders you unable to speak by actively breaking your jaw! Most setters are peaceable sorts after all.
24 Yoghurty drink US hacker smuggled into satellite (7)
MADZOON – ADZ in MOON. This is popular in Armenia, apparently. My first crack at an answer to this clue resulted in IAXMANO, which sounds tasty to me but is sadly not actually found anywhere.
25 Beside Devon river, nine shrubs needing little water (7)
TAMARIX – TAMAR + IX. Trust the wordplay!
26 Sets out anew, going over mountain face in second year (10)
REDISPLAYS – reversed ALP SIDE [mountain | face] in S YR
27 Men from S Africa’s classic race given voice (4)
OKES – homophone of OAKS, a flat horse race run at Epsom Downs, maybe even a race in Kentucky too.
DOWN
1 Such is zloty, pay with secondary deductions leads to mild disorder (10)
SCHIZOTYPY – S{u}CH I{s} Z{l}OTY P{a}Y. An inventive cluing device!
2 Examination to check stray civet (7)
VIVERRA – VIVA “checks” ERR. Again, trust in the straightforward wordplay!
4 Performed badly, cut up about pet, one producing stink (9)
EFFLUVIUM – MUFFE{d} reversed about LUV I [pet | one]
5 Turn weak, entering constant darkness up north (5)
NICHT – reverse THIN, then enter C [the immutable speed of light]
6 Short slip with US volume on radical Islam’s Semitic feature (13)
TRILITERALISM – TRI{p} + LITER + (ISLAM*). So this is a feature of Semitic languages where words have roots of three consonants. (Trust the wordplay.)
7 “Vital energy connected to river/sea”, local tweeted (7)
CHIRMED – CHI + R MED. Not CHIRRED as I originally essayed, but fortunately there was a helpful crosser here. My dictionary app doesn’t actually list this as regional, but it’s certainly not that normal, is it?
8 Top-to-bottom change in personality needed for Shakespearean tangles (4)
ELFS – Take SELF [personality] and put the top letter to the bottom. Think elf-locks.
10 Safe inversion of Sondheim show? (3,2,3,5)
OUT OF THE WOODS – The Sondheim musical being, of course, “Into The Woods”.
13 Experiencing pain when wearing undergarment with raised boundary and short line of symmetry (10)
BRACHYAXIS – I saw ACHY in BRA at once, but needs some crossers to see that “raised boundary” was not MIR or MEH but a reversed cricketing SIX.
16 Composition of doggerel from Old French and in Danish, say (9)
POETASTRY – O ET [old | French “and”] in PASTRY
18 Fresh university location for Newark and even parts of Land’s End (7)
UNJADED – U + NJ [location for (America’s) Newark] + {l}A{n}D{s} E{n}D
20 UK right to spaff money overseas (7)
TUGHRIK – (UK RIGHT*). As all quiz list-learners will know, the tug(h)rik is the currency of Mongolia and made up, somewhat mercilessly, of 100 mongos.
22 No end of interest in payment for use of an organ (5)
RENAL – REN{t}AL [payment for use], subtracting the last letter of {interes}T.
23 Perth’s wet weather in which spirits must be kept up (4)
SMUR – reversed RUMS. As so often in these things, this is Perth in Scotland, not the Antipodes.
That process was especially helpful with FERNYTICLE, which according to Chambers has no less than 10 alternative spellings, only one of which actually worked.
Kudos to the setter for even attempting to clue SCHIZOTYPY, let alone producing an almost readable surface.
Worth doing.
And three comments already! Horrid will be pleased!
Edited at 2020-05-01 11:29 am (UTC)
The minutes –
We have ordered more SCHIZOTYPY from IKEA Budapest and have received warm wishes from Sir Walter FERNYTICLE QC (Kelso)
At 1ac it is noted that AVIS (Bombay) – the car hire people – sponsored this clue – is that allowed?
I think that’s it oh! Zed’s just turned up.
Air taxi for one – thank-you Phil – Meldrew