The Asari are one of the iconic races from the popular Mass Effect games and universe, and while the image above isn't from any official media, it's close enough to what they look like. Besides their biotic abilities, which are fantastic enough to be disregarded here, they are a mono-gender race, reproducing by "melding" with another sentient organic and randomizing their own DNA using the partner as a template. How/why might such a reproductive mechanism evolve, and how might it work?
Step 1: An asari ancestor species had normal sexual reproduction, the species being hermaphroditic. Reproduction was through meetings of two gametes, but it wasn't through mixing of DNA as happens on Earth; the egg contained all the genetic information available to form a zygote, but in a somewhat disorganized state. The sperm-analogue carried protein catalysts that caused the egg's genetic material to organize, with minor variations in the final genetic code based on slight differences in the protein catalysts. When the zygote matured, it produced slightly altered catalysts due to its slightly altered genetics compared to the "female" parent.
In essence, genetic traits are passed on not by mixing material from both parents, but by passing on how the material from one parent (the "mother") is organized by the other (the "father").
The proto-asari also have organelles that can detect very weak electromagnetic fields, and can produce their own electricity to sense other life forms (as some species of fish can). They use this like other animals use scent to identify potentially compatible and fertile mates, and mutual EM stimulation becomes a part of proto-asari reproduction.
Step 2: Parthenogenesis takes place in some individuals. They no longer need to mate, and can use the protein catalysts they produce to "fertilize" their own eggs. However, as the catalysts are produced using their own genetic information, a fertilized egg's genetics are identical to that of the parent (excepting the odd mutation). Just like some parthenogenetic lizards and fish require simulated sex in order to reproduce, the proto-asari still need the EM stimulation in order to trigger reproduction.
Step 3: Parthenogenetic individuals evolve the ability to epigenetically alter their production of protein catalysts. This alteration gets tied into the proto-asari's EM sense. The proto-asari re-invent sex. They can use the weak electromagnetic field produced by another proto-asari to affect the proteins catalysts when they self-fertilize. Without the need to physically transfer chemicals from one to another, the "male" components of the physiology are eventually lost.
Step 4: Eventually, the asari are found by the Protheans. The Protheans not only alter them in order to give them biotics, they tweak them so that asari reproduction can be stimulated by the EM fields generated by the brains of other intelligent life forms, and give them some unconscious control over protein catalyst synthesis. When the stimulation takes place, the asari unconsciously produces the protein catalysts that will replicate something of what they perceive in the other species.
I just realized a different model; in advanced Thessian species, they have a mitochondrial-like organelle with its own genetic material and coding that instead of being involved in energy production is involved in the transcription of the cellular DNA-analogue during duplication. Original reproduction involved the "male" providing the organelle which entered the egg and allowed mitosis to take place. It's differences within the organelle's genetics that cause differences in the way it transcribes the cellular genetics, causing variations to arise. Everything else in the proposed evolution proceeds as I described, with Step 3 being the epigenetic alteration of the organelle's genes.
"Why might such a reproductive mechanism evolve?"
This belies a fundamental misunderstanding of evolution. It replaces Evolution with Lamarckism, a key competitor with Evolution until Science managed to collect enough evidence that Lamarck's theory wasn't supported by observation of the world.
Evolution is the apparent adaptation of organisms, where adaptation is not driven by a purpose; but is a leftover of the organisms that adapted in non-successful ways dying out.
How might it work?
It depends. Being a fictional character, one is not bound to real world observations based in known Science. But, there are real world reproductive oddities, that we might draw inspiration from.
- Yeast / Fungus
These animals merge their cell walls together, leading to multi-nucleated cells where each individual nucleus carries a copy of DNA.
Bacteria can "conjugate" transferring one's replicated DNA into another's cell. This conjugation may lead to a hybrid organism that is genetically distinct from both original organisms.
These types of organisms carry both sets of gametes, such that they are technically both male and female simultaneously. Typically they are protected against self-procreation, and sex occurs normally, with one or both organisms becoming pregnant.
- Assisted genetic recombination
Viruses carry genetic payloads that become incorporated into the cell's protein production pipeline. For some of the viruses, those payloads include helper proteins that splice the DNA into the organisim's own DNA.
- Plasmid DNA transfection
Plasmids can deliver their DNA to a cell, infecting it with foreign DNA. The process is known as transfection.
- Environmental uptake
Small cellular organisms sometimes pick up bits of DNA and incorporate them internally without breaking it down to basic Nucleic Acids.
How might it work (round two)
Since the mechanism is described as "rearranges their own DNA according to another's template" it is unclear if reproduction took place. Normally, to reproduce, you clearly have a second organism at one point in time. An organism that rearranges it's own DNA hasn't produced a second organism.
Assuming that this "rearrangement" is compartmentalized to a portion of the organism which is destined to become a new organism, you have a near standard formula for sexual reproduction.
You see, it's not the presence of DNA that is significant in Biological organisms. It is the information encoded within the DNA. Due to some ancient event, we carry around additional copies of what we need to survive. Sexual reproduction is where our offspring lose (at least) one of the copies and gains a (likely) new copy. These copies are inexact, which gives sexual organisms a faster route to reordering their information than non-sexual organisms.
Depending on what is being rearranged, it is unclear to me if this organism would even need special consideration to fit in as a rank-and-file biological hermaphrodite.
How might it work? (round three)
Our current understanding of DNA is tightly tied to it's replication and production of mRNA which is further transcribed into proteins. This occurs with linear DNA, where parts of it actively create proteins.
Assuming we take "rearrangement" to not mean rearrangement of alleles (the possible alternate variations found at a gene site) but rearrangements of the DNA itself, then the organism might
have very tiny genes, such that rearrangement never splits a gene required for survival. Even then it's a statistics game, large number of combinations would result in dead offspring.
have a mechanism where genes are rearranged but in ways that are guaranteed to not be split. This has some interesting promise, but the discovery of "homeobox genes" or genes that activate and deactivate regions of other genes would not work properly, likely leading to a dead organism.
homeobox genes are partially responsible for making sure your muscles are comprised by tissue that's muscle instead of bone, blood, nerves, hair, skin, or some other tissue. Genes jumping into and out of homeobox regions would have as big an impact on the organism's growth as genes being split into two non-properly working regions.
have a mechanism that assured that regions controlled homeobox genes were not rearranged. This idea sounds great, but is very unlikely. It basically presupposes that the organism being born knows how to avoid being born with birth defects. No organism can decide a proper course before it exists.
have a modular inter-celluar structure, such that the location of the gene doesn't impact it's utility. Sounds great, but just ignores the gene getting split into two nonfunctional pieces by "begging the question" or "assuming a scenario where the failure can't occur because you've defined the scenario to exclude the failure conditions"
How might it work? (final round)
Currently history, which includes luck and chance, has shown that typically sexed organisms dominate the larger species. That is because they are more effective in having offspring that has a wider range of DNA recombination. This wider range permits more members of the species to survive.
We have natural hermaphrodites in nature today. For one reason or another, the chance for them to dominate the environment has not been realized. Sex based organisms are more successful in the Animal and Plant based kingdoms. It might be due to luck; but, it may also be that any specific organism that doesn't have to carry around two sets of reproductive organs has an advantage in living to adulthood.
The Asari genome is an aftermarket add-on.
Here is the Atmosphere Demolisher spoiler, featured by a number of different model cars. Imagine if the possession of a bigass spoiler like this made the car an Asari car, regardless of the make or model. I propose that is how the Asari genome works.
It is not an outrageous proposition. The Y chromosome is basically an aftermarket add-on that turns the base human model (female) into a male. It makes for interesting reading, the Y. https://en.wikipedia.org/wiki/Y_chromosome
The Y chromosome is tiny and degenerate, representing 2% of total DNA in a human but it is the sole difference between male and female human.
So too your Asari. The tiny Asari genome is an aftermarket add-on. Their reproductive strategy aims at sentient species, and modifies the body plan of that species with a 2% add-on chromosome that makes that individual an Asari.
I gather that in the game, Asari always look like sexy space chicks the males are digging. But the proposed reproductive strategy here would be a lot easier if the other-species mate were a female rather than a male. The biological apparatus for producing children might vary a lot from species to species and a "female" Asari from stock species A might not have the hardware to produce young with mate species B. The whole male thing is to donate some genetic material to something capable of having your children; this strategy would work well with the Asari too.
They could still look like sexy space chicks if you are into that kind of thing. They still might shake it for the males. Reproduction, however, is a different matter...