Hints:
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1) Understand the full parity, +/-3, and central X-wing properties of non-consecutive 6x6s.
2) solve algebraically, using letters as placeholders.

Stated without proof:
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In a 6x6 with the non-consecutive rule, these properties *always* apply:
(A) The boxes divide into an odd layer and an even layer. Obviously, this causes roping.
(B) The abutt odd-odd and even-even vertically (so is OEEOOE one one side and EOOEEO on the othe)
(C) Ever vertical domino in a box is X with X+/-3
(D) The central 2 columns are three X-wings.

Proof isn't difficult, but it should be trivial to see that everything that does satisfy the above will automatically satisfy the non-consecutive constraint:
- rows are always separated by 3 (O<->E) or 2 or 4 (O<->O or E<->E), and
- columns always separated by 2 or 4 (within box) or 3 (across the middle box boundary)

The solve:
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Label r4c2 'A', we don't know its value, but later letters are higher values until they wrap back to 1 again.
By (C), r3c2 must be D (being A+3), therefore r3c3 is B or F.
By (D), r4c4 is the same as r3c3, so B or, if A=1, also F
By (B), r5c6 is the same parity as r4c[456]
By thermo, r5c6 is not the same as r4c5
By sudoku, r5c6 is not the same as r4c6
=> r5c6 is the same as r4c4, and therefore r3c4, so also B or conditionally F
By thermo r4c5 is greater than r5c6, and given the assumption that F hasn't wrapped round, so A<F, r3c3, r4c4, and r5c6 cannot be F and are B.
=> r4c5 is D or F

If A is 4 or above, D and F will have wrapped to be less than B, breaking the r5c6 thermo
If A is 3 or 2, and D is 6 or 5 and are still available, but F will have wrapped to be less than B
If A is 1, both D and F remain possibilities.

By (A), given that r4c[123] are C-A-E, or {A+2}-A-{A+{4,-2}}, r5c2:=r5c1+{4,-2}
By (C), and r6c2:=r5c2+3:=r5c1+{1,-5}
By thermo, for r6c2:=r5c3+{1,-5} to be *less* than r5c3, that +1/-5 must have caused the wrap from 6 to 1
=> the r6c2 thermo is 1-6, and the 6 in row 4 is in r4c3, the E
=> A=2, ... E=6, F=1
=> by thermo, r4c5 is D not F

Following banding, the rest of the cells fill immediately.

Proof of uniqueness:
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https://sigh.github.io/Interactive-Sudoku-Solver/?q=.Shape%7E6x6.AntiConsecutive.Thermo%7ER6C2%7ER5C1.Thermo%7ER4C2%7ER3C3.Thermo%7ER5C6%7ER4C5

Side note - this is a similar puzzle but with a simpler solution:
https://sigh.github.io/Interactive-Sudoku-Solver/?q=.Shape%7E6x6.Thermo%7ER2C2%7ER1C3.Thermo%7ER3C6%7ER2C5.Thermo%7ER5C1%7ER4C2.AntiConsecutive