Halliday & Matthiessen (2014: 614-5, 616n):
Certain items that serve as ‘elaborating’ conjunctions that are ‘internal’ only in orientation may also serve as modal adverbs (e.g. actually, in fact, indeed, as a matter of fact), functioning either as mood Adjuncts of intensity or as comment Adjuncts of factuality. In fact, these are related historically through processes of grammaticalisation: starting from an experiential source, items such as these tend to develop into textual conjunctions, and by a further step into interpersonal modal adverbs…
Some of the items that are used as conjunctions with verifactive senses also have assessment senses and serve as mood or comment Adjuncts: actually, in fact, indeed – (i) verifactive (‘in reality’), (iia) intensity: counterexpectancy (‘even’), and (iib) factual (speech functional comment: ‘really’); as a matter of fact – (i) verifactive (‘in reality’), and (ii) factual (speech functional comment: ‘really’). The link between the verifactive and assessment senses is the orientation towards the interpersonal: in their assessment senses, these items are, of course, purely interpersonal; and in their verifactive senses, they are internal in orientation (cf. Halliday & Hasan, 1976: 240ff.): the clarification relates to the speech function rather than to the experiential content of the proposition or proposal being exchanged – ‘I’ll tell you by way of clarification of what I said before’. Thus in the following example, in fact serves as a cohesive conjunctive Adjunct and actually as a mood Adjunct: But that basic training helped me in the part. In fact, when Tom Quayle (Tom Jennings) actually did knock me down in the fight scene, I saw red and had to take a few deep breaths and hold myself back. Traugott (1997) shows how both indeed and (during a later period in the history of English) in fact developed (in our terms) from items serving as circumstantial Adjuncts, to items serving as interpersonal Adjuncts and also to items serving as conjunctive Adjuncts. They follow one of the grammaticalisation paths she has identified.