To my knowledge (Please correct me if I am wrong), genetic basis is the key in defining species. When we encounter an unknown species, we can sequence it's genome and compare the genome with other known organisms from the genome data bank and see if there is a significant similarity. If no significance were found then we can call it a newly discovered organism. Also careful study of genetic crosses between species pairs and partially isolated sub species to identify general patterns in the genetics and origin of reproductive isolation helps to answer the question of speciation. Then how would we cross asexual organisms like many bacteria or protists? Is it possible to cross asexual organisms? If not, how do scientists study their genetics and define species of asexual organisms?
When it comes to viruses and bacteria, genetic cross is just as a vaguely defined concept as species. The latter is often defined on the percentage of the sequence similarity (for viruses) or the percentage of common genes (for bacteria).
If we speak about simply mixing genomes of different organisms, there are many possibilities, how this could happen. Just to give a couple of examples:
- Some viruses integrate their genome into the host genome. In some cases these integrated genomes cease being functional viruses, and may evolve to be functional genes of the host. This is truly a genetic cross. See here, for example.
- Bacteria and viruses routinely undergo horisontal gene transfer in which genes from one bacteria are integrated into the genome of the other one. A typical bacterial species has a few hundred core genes, and thousands of genes that are present only in some specimen or borrowed from other species, which significantly vary from one bacteria to another. Thus, every individual bacteria or bacterial community can be thought of as a genetic cross.