  Native language identification (NLI) is the task of training (via supervised
machine learning) a classifier that guesses the native language of the author
of a text. This task has been extensively researched in the last decade, and
the performance of NLI systems has steadily improved over the years. We focus
on a different facet of the NLI task, i.e., that of analysing the internals of
an NLI classifier trained by an \emph{explainable} machine learning algorithm,
in order to obtain explanations of its classification decisions, with the
ultimate goal of gaining insight into which linguistic phenomena ``give a
speaker's native language away''. We use this perspective in order to tackle
both NLI and a (much less researched) companion task, i.e., guessing whether a
text has been written by a native or a non-native speaker. Using three datasets
of different provenance (two datasets of English learners' essays and a dataset
of social media posts), we investigate which kind of linguistic traits
(lexical, morphological, syntactic, and statistical) are most effective for
solving our two tasks, namely, are most indicative of a speaker's L1. We also
present two case studies, one on Spanish and one on Italian learners of
English, in which we analyse individual linguistic traits that the classifiers
have singled out as most important for spotting these L1s. Overall, our study
shows that the use of explainable machine learning can be a valuable tool for
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