A standard format used for storing the output of high-throughput sequencing experiments is the FASTQ format. It comprises three main components: (i) headers, (ii) bases (nucleotide sequences), and (iii) quality scores. FASTQ files are widely used for variant calling, where sequencing data are mapped into a reference genome to discover variants that may be used for further analysis. There are many specialized compressors that exploit redundancy in FASTQ data with the focus only on either the bases or the quality scores components. In this paper we consider the novel problem of lossy compressing, in a reference-free way, FASTQ data by modifying both components at the same time, while preserving the important information of the original FASTQ. We introduce a general strategy, based on the Extended Burrows-Wheeler Transform (EBWT) and positional clustering, and we present implementations in both internal memory and external memory. Experimental results show that the lossy compression performed by our tool is able to achieve good compression while preserving information relating to variant calling more than the competitors. Availability: the software is freely available at https://github.com/veronicaguerrini/BFQzip.