Motivation: Word-based or 'alignment-free' methods for phylogeny reconstruction are much faster than traditional approaches, but they are generally less accurate. Most of these methods calculate pairwise distances for a set of input sequences, for example from word frequencies, from so-called spaced-word matches or from the average length of common substrings. Results: In this paper, we propose the first word-based approach to tree reconstruction that is based on multiple sequence comparison and Maximum Likelihood. Our algorithm first samples small, gap-free alignments involving four taxa each. For each of these alignments, it 1 arXiv:1803.09222v2 [q-bio.PE] 28 Apr 2018 then calculates a quartet tree and, finally, the program Quartet Max-Cut is used to infer a super tree topology for the full set of input taxa from the calculated quartet trees. Experimental results show that trees calculated with our approach are of high quality. Availability: The source code of the program is available at https://github.com/tdencker/multi-SpaM