Halliday & Matthiessen (1999: 25):
We have retained the term 'realisation' to refer to the interstratal relationship between the semantics and the lexicogrammar: the lexicogrammar 'realises' the semantics, the semantics 'is realised by' the grammar. … In any strata! system (i.e. any system where there are two strata such that one is the realisation of the other) there is no temporal or causal ordering between the strata. It makes no sense to ask which comes first or which causes which. That would be like taking an expression such as x = 2 and asking which existed first, the x or the 2, or which caused the other to come into being (it is not like the chicken and the egg, which are temporally ordered even though in a cycle). There is a sense in which realisation is the analogue, in semiotic systems, of cause-&-effect in physical systems; but it is a relationship among levels of meaning and not among sequences of events, … the relationship is an intensive one, not a causal circumstantial one.
Blogger Comments:
This unexplained analogy is potentially misleading, given that the identifying relation between strata is intensive (elaborating), not circumstantial (enhancing: causal).
One possible explanation is that, as an identifying relation, if semantic values are encoded by reference to lexicogrammatical tokens, the lexicogrammar serves as the agent (causal participant) of the identifying process, with semantics as the medium through which the process unfolds. And this is essentially Halliday & Matthiessen's theoretical method: construing a semantics by reference to the lexicogrammar.
However, whether this is what Halliday & Matthiessen mean by this unexplained analogy is anyone's guess.