Having eaten lunch, I went out.
Is it correct and grammatical to say
I went out having eaten lunch.
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I believe that 'having eaten lunch' is, here, a descriptive gerund phrase. As such, it can appear in either position, though some people might find it easier to read with an added comma, especially if it were part of a longer sentence:
I went out, having eaten lunch, and street-food sellers immediately assailed me with their cries.