What Does The Theoretical Term 'Realisation' Actually Mean Precisely?

Linguistic theory is 'language turned back on itself'. This just means that linguistic theory is the use of language to model language.

As a consequence, the language of theory can be analysed in terms of the theory of language.

For example, the theoretical relation of 'realisation' can be understood in terms of grammatical theory.

In transitivity theory, 'realise' is an intensive identifying Process — subtype 'symbol'.

It construes an identity between two levels of symbolic abstraction: a lower Token and a higher Value.

In an identity relation, either level can be used to identify the other.

If the lower Token is used to identify the higher Value, the relation is encoding:

the identity encodes the Value by reference to the Token.

If the higher Value is used to identify the lower Token, the relation is decoding:

the identity decodes the Token by reference to the Value.

So when the language of theory says that the stratum of wording (lexicogrammar) realises the stratum of meaning (semantics), the theory of language says that this construes an identity between two levels of symbolic abstraction, a lower Token (wording) and a higher Value (meaning).

If the lower Token (wording) is used to identify the higher Value (meaning), then the identity encodes the meaning by reference to the wording.

If the higher Value (meaning) is used to identify the lower Token (wording), then the identity decodes the wording by reference to the meaning.

(In this regard, Halliday's argument in On the ineffability of grammatical categories was that a theory of meaning can only be encoded by reference to a theory of wording.  That is, a theory of wording cannot be decoded by reference to a theory of meaning.)

Similarly, when the language of theory says that form (e.g. verbal group) realises function (e.g. Process), the theory of language says that this construes an identity between two levels of symbolic abstraction, a lower Token (form) and a higher Value (function).

If the lower Token (form) is used to identify the higher Value (function), then the identity encodes the function by reference to the form.

If the higher Value (function) is used to identify the lower Token (form), then the identity decodes the form by reference to the function.

Finally, when the language of theory says that structure (syntagmatic axis) realises system (paradigmatic axis), the theory of language says that this construes an identity between two levels of symbolic abstraction, a lower Token (structure) and a higher Value (system).

If the lower Token (structure) is used to identify the higher Value (system), then the identity encodes the system by reference to the structure.

If the higher Value (system) is used to identify the lower Token (structure), then the identity decodes the structure by reference to the system.