BQN, 9 bytes

≡⟜∧·∾´·∧⊔ 

Anonymous tacit function. Takes a list of integers. Try it at BQN online!

Explanation

The secret sauce here is the ⊔ Group builtin. In a monadic context, it takes the indices of a list and divides them up into different "buckets" based on the value at each index. For example, ⊔ ⟨4,4,1,2,1⟩ gives ⟨⟨⟩,⟨2,4⟩,⟨3⟩,⟨⟩,⟨0,1⟩⟩: there are no 0s; there are 1s at indices 2 and 4; there is a 2 at index 3; there are no 3s; and there are 4s at indices 0 and 1.

If all occurrences of the same digit are consecutive, then sorting these groups of indices will cause all the indices to be in ascending order. Otherwise, some indices will be out of order even after sorting their groups.

≡⟜∧·∾´·∧⊔   ⊔ Group indices of the argument list ·∧ Sort ascending (in lexical order) ·∾´ Fold on concatenate (flatten) ≡⟜  The result is identical to ∧  The result, sorted 

BQN, 9 5 bytes

≡⟜∧⊐˜ 

Anonymous tacit function. Takes a string or a list of integers. Try it at BQN online!

Explanation

Inspired by Dennis's J answer, though the details are a bit different.

Given two lists, ⊐ works as follows: for each element of its right argument, it finds the first index of that value in its left argument.

We're using ⊐˜, which passes the same list as both left and right arguments. The result is a list, for each digit, of the index of that digit's first occurrence.

If the digits are grouped together, these indices will occur in ascending order. However, if some copies of digit A are separated by digit B, B's (larger) index will come between the copies of A's (smaller) index.

≡⟜∧⊐˜ ⊐˜ First index of each digit in the argument list ≡⟜ The result is identical to ∧ The result, sorted ascending