NC increasing (ONC), 89 not allowed, 98 allowed.
Prelims
a) R9C89 = {18/27/36}/[54] (ONC), no 9
b) 26(4) cage at R1C1 = {2789/3689/4589/4679/5678}, no 1
c) 27(4) cage at R7C8 = {3789/4689/5679}, no 1,2
d) 14(4) cage at R8C6 = {1238/1247/1256/1346/2345}, no 9
1a. 45 rule on R12 2 innies R12C9 = 6 = {15/24}
1b. 45 rule on C89 using R12C9 = 6, 2 outies R12C7 = 6 = {15/24}
1c. Naked quad {1245} in R12C79, locked for N3
1d. R12C7 = 6 -> R12C8 = 9 = {36}, locked for C8 and N3, clean-up: no 3,6 in R9C9
1e. Naked triple {789} in R3C789, locked for R3
1f. 45 rule on N9 3 innies R789C7 = {135/234} (cannot be {126} which clashes with R12C7), 3 locked for N9
1g. Naked quint {12345} in R12789C7, locked for C7
1h. 27(4) cage at R7C8 = {4689/5679}
1i. R9C89 = {18/27} (cannot be [54] which clashes with 27(4) cage), no 4,5
1h. 6 in C7 only in R456C7, locked for N6
1i. Min R3C89 = 15 -> max R4C89 = 7, no 7,8,9
1j. Min R34C7 = 13 = [76] -> max R34C6 = 7 but cannot be {16}, no 6,7,8,9
2a. 45 rule of C6789 using R12C9 = 6, 3 outies R127C5 = 9 = {126/135/234}
2b. 45 rule of C5 using R127C5 = 9, 2 innies R89C5 = 11 = {29/38/47}/[65] (ONC), no 1
2c. R89C5 = 11 -> R89C4 = 9 = {18/27/36}/[54] (ONC), no 9
2d. 45 rule of C6789 using R12C9 = 6, 2 innies R12C6 = 1 outie R7C5 + 14
2e. Min R12C6 = 15 = {69/79}/[87/98] (ONC)
2f. Max R12C6 = 17 -> max R7C5 = 3
2g. R127C5 = {126} can only be [621] (because [261] clashes with R12C6 = [87], ONC), no 6 in R2C5
2h. R12C6 = {79}/[87/98] (cannot be {69} because R7C5 = 1, R12C5 = {35} and 23(4) cage at R1C5 = [3659/5936] (ONC) clashes with one of R12C8), no 6
2i. 25(4) cage at R3C5 = {1789/2689/3589/3679/4579/4678}
2j. 1 of {1789} must be in R3C5 -> no 1 in R456C5
2k. Max R7C57 = [35] = 8 -> min R7C6 = 5 (because {345} must be [354], ONC)
3a. 45 rule on R89 4(3+1) innies R8C789 + R9C1 = 28, max R8C789 = 24 -> min R9C1 = 4
3b. 45 rule on R89 4(2+2) innies R89C1 + R8C89 = 28, max R8C89 = 17 -> min R89C1 = 11, no 1 in R8C1
[I ought to have spotted this earlier; I never used it when solving the simpler version.]
4a. 45 rule on N89 1 innie R8C4 = 8, clean-up: no 1 in R89C4 (step 2c), no 3 in R89C5 (step 2b)
4b. R8C4 = 8 = R7C34 + R8C3 = 10 (127/136/145/235}, no 9, no 7 in R7C4 + R8C3 (ONC)
4c. 8 in N2 only in R12C6 = [87/98] (ONC) = 15,17 -> R7C5 = {13} (step 2d)
4d. 12(3) cage at R7C5 = {129/147/156/237}/[354] (cannot be {246} because R7C5 only contains 1,3)
4e. R7C5 = {13} -> no 1,3 in R7C7
4f. 3 in N9 only in R89C7, locked for 14(4) cage at R8C6
4g. R89C4 + R89C5 cannot be {27}[65] which clashes with R127C5 = {35}1/[621] and with 12(3) cage at R7C5 = [354/372] -> no 2,7 in R89C4
4h. R89C5 = {29/47} (cannot be [65] which clashes with R89C4), no 5,6
4i. 12(3) cage = {129/147/156/237} (cannot be [354] which clashes with R89C4)
4j. 14(4) cage at R8C6 contains 3 = {1346/2345}, no 7
4k. 1 of {1346} must be in R89C6 (R89C6 cannot be {46} which clashes with R89C4), no 1 in R89C7
4l. 1 in C7 only in R12C7 (step 1b) = {15}, locked for N3, 5 locked for C7
4m. Naked pair {24} in R12C9, locked for C9
4n. Naked triple {234} in R789C7, 2,4 locked for N9, clean-up: no 7 in R9C89
4o. Naked pair {18} in R9C89, locked for R9, 8 locked for N9
4p. 12(3) cage = {129/147/237} (cannot be {156} because 5,6 only in R7C6) -> R7C6 = {79}
4q. Killer pair 7,9 in R12C6 and R7C6, locked for C6
4r. R12C6 + R7C5 (step 2d) = [871/983] -> 12(3) cage = [192/372] (cannot be [174] which clashes with R12C6 + R7C5) -> R7C7 = 2, R89C7 = [43] (ONC), no 2 in R9C6 (ONC) -> R89C6 = 7 = [16/25], clean-up: no 6 in R8C4 (step 2c), no 7 in R9C5 (step 2b), no 5 in R8C7 (ONC)
4s. 7 in R9 only in R9C123, locked for N7
4t. R127C5 (step 2a) = {126/135} (cannot be {234} which clashes with R89C5), no 4, 1 locked for C5
4u. 2 of {126} must be in R2C5 -> no 2 in R1C5
4v. 4,6 in N8 only in R9C456, locked for R9
5a. 45 rule on C34 using R89C4 = 9 (step 2c) 4(2+2) innies R5C34 + R89C3 = 27, max R5C34 = 16 (cannot be [89], ONC) -> min R89C3 = 11, no 1
5b. Min R89C3 = 11 -> max R89C2 = 8, no 8,9
5c. Min R89C2 = 5 (cannot be [12], ONC) -> max R89C3 = 14 -> min R5C34 = 13, no 1,2,3
6a. R89C1 + R8C89 = 28 (step 3b), min R8C89 = {79} = 16 -> max R89C1 = 12, no 2
6b. 2 in N7 only in 19(4) cage at R8C2 = {1279/2359} (cannot be {2368} because 3,6,8 only in R8C23), no 6,8, 9 locked for C3 and N7
6c. R8C1 = 8 (hidden single for N7)
6d. R8C9 = 6 (hidden single in R8) -> no 5 in R7C9 (ONC)
6e. R7C8 = 5 (hidden single in N9) -> no 4 in R6C8 (ONC)
[With hindsight, if I’d spotted step 9a next, step 8 and particularly 8a would have been simpler.]
7a. Variable hidden killer pair 4,6 in R7C12 and R7C3 for R7 -> R7C12 must contain at least one of 4,6
7b. 15(4) cage at R6C1 = {1248/1347/2346} (cannot be {1239/1257/1356} which don’t contain 4 or 6), no 5,9
8a. 22(4) cage at R3C8 = [7915/8725/8743] (cannot be {79}{24} because 2,4 only in R4C8, cannot be [9823] (ONC), cannot be [9715] because R89C8 cannot be [78] (ONC), cannot be [9841] which clashes with R9C9), no 7 in R3C7, no 9 in R3C8, no 8 in R3C9, no 1 in R4C9
8b. 7 in C7 only in R456C7, locked for N7
8c. R3C89 = [79]
or R3C89 = [87] => R7C9 = 9
-> no 9 in R56C9
8d. 17(4) cage at R5C8 = {1259/1349/2348} (cannot be {1358} which clashes with R4C9)
8e. 1,3,5 of {1259/1349} must be in R56C9, 3,8 of {2348} must be in R56C9 -> no 1,8 in R56C8
8f. {2348} must be [4328] (ONC) -> no 8 in R5C9
8g. Min R34C6 = 4 (cannot be [21] which clashes with R8C6) -> max R34C7 = 16, no 8,9 in R4C7
8h. Min R34C7 = 14 -> max R34C6 = 6 = {13/14/24}/[32] (ONC) (cannot be {15} which clashes with R89C6), no 5
8i. Killer pair 1,2 in R34C6 and R8C6, locked for C6
9a. Consider combinations for 19(4) cage at R8C2 (step 6b) = {1279/2359}
19(4) cage = {1279} => R9C1 = 5 (hidden single in N7)
or 19(4) cage = {2359}, 3 locked for R8 => R8C4 = 5
-> R9C6 = 6, R8C6 = 1 (cage sum), R7C5 = 3 -> R7C6 = 7 (cage sum)
9b. R12C6 = [98] = 17 -> R12C5 = 6 = {15}, locked for N2, 5 locked for C5
9c. R7C9 = 9, R8C8 = 7, R3C789 = [987], R9C89 = [18]
9d. R89C4 = [54]
9e. Naked pair {29} in R89C5, locked for C5
9f. 4 in N2 only in R3C56, 4 locked for R3
9g. 7 in N2 only in R12C4, locked for C4 and 20(4) cage at R1C3
9h. 19(4) cage = {2359}, 5 locked for N7 -> R9C1 = 7
9i. 1,5 in R3 only in R3C123, locked for N1
9j. 15(4) cage at R6C1 (step 7b) = {1248/1347/2346}, R7C12 = {14/46} -> R6C12 = [28/37/32] (ONC) -> R6C1 = {23}, R6C2 = {278}
10a. R5C34 + R89C3 (step 5a) = 27
10b. R5C4 = {69} -> R589C3 must contain 9 in R89C3 = 18,21 = {279/369/459/579}
10c. 4,6,7 only in R5C3 -> R5C3 = {467}
11a. 18(4) cage at R6C3 = {136}8/{145}8 (cannot be {235}8 because R7C3 only contains 1,4,6, cannot be [7218] which clashes with R6C12), no 2,7
11b. 18(4) cage = {13}[68]/[5148] (cannot be {36}[18] which clashes with 15(4) cage at R6C1), 1 locked for R6, no 1 in R7C3
11c. 1 in R7 only in 15(4) cage at R6C1 (step 7b) = {1248/1347} -> R7C12 = {14}, R7C3 = 6, R6C34 = {13}, 3 locked for R6, R6C12 = [28], R6C89 = [95] = 14 -> R5C89 = 3 = [21], R4C89 = [43]
11d. R6C6 = 4, R5C6 = 5 (hidden single in C6) -> R56C7 = 15 = [87] -> R3456C5 = [4876], 20(4) cage at R3C6 = [3926], R456C4 = [193], R56C3 = [41]
11e. R4C3 = 7 (hidden single in C3), R4C4 = 1 -> R3C34 = 8 = [26], R2C3 = 6, R2C8 = 6 -> R2C7 = 1 (ONC), R1C8 = 3 -> R1C9 = 2 (ONC)
and the rest is naked singles without using ordered NC.