Once one considers carefully what is meant by "life after death", it is very difficult to avoid the realization that if it exists at all, it must be exceedingly rare or else leave virtually no detectable trace whatsoever "in this world".
Follow with me.
In the literal sense, sure, there is life after death. Hardly anyone disputes that people die, yet life goes on. Nor is the issue whether there can be some period of consciousness after the heart and lungs have stopped.
While there those are who believe in the literal ressurrections of the body in this world, it seems fairly well-established that even the most extreme forms of such a belief accept that it is exceedingly unusual in both the past and the present. Perhaps the ancient Egyptians believed differently; I am not sure. I will leave that aside for now.
So I think we can agree that for the purposes of this thread the question in full is whether it is possible and/or usual for people who have gone through irreversible brain death to somehow retain or renew consciousness after death, either as some form of astral body, through reincarnation, or perhaps in a whole separate realm that can not usually be perceived from this world. Or some combination of those scenarios, perhaps.
Assuming that we can agree on that hypothesis, it seems to me that it almost completely blocks any possibility of actual scientific evidence, although it has been attempted nonetheless. Claims of the paranormal would be necessary for all the hypothetical situatins except reincarnation, and such claims can hardly ever be falsifiable, mostly because there are no means to establish that they are genuine and correctly interpreted.
That leaves us with attempts to establish that reincarnation exists. While there are those who do not hesitate to say that it has been done, mostly by referring to the work of Ian Stevenson and to claims about Tibetan, Egyptian or Japanese beliefs, that is tentative at the very best, and vanishes entirely once one considers the logical consequences of actual reincarnation.
To witness, reincarnation claims are at their core appeals to the belief that people "come back in another body" after dying, often by proposing that there is some form of immortal soul that is the person's "true self".
Such a claim, of course, is impossible to disprove. A literal infinity of claims is similarly impossible to disprove, many of those being ludicrous and/or meaningless. But we don't need to go there. Let's instead consider the practical consequences if meaningful reincarnation existed.
One possible scenario, one that by my understanding is actually held as true by many Hindus, is that reincarnation does not involve memories of past lifes to any significant degree, even if the soul/Atman is presumably the very same. Far as I can see it is impossible to evidence for or against such a hypothesis. For all practical purposes, such a scenario is completely indistinguishable from what one would expect from a "materialistic" view. People die. People are born. The world that they exist at is influenced by their existence in various ways. And after that it is mostly in the eyes of the beholder. So it is a resilient view, but not really a demonstrable one.
A slightly bolder proposal would go along the lines of traditional Tibetan beliefs, which say in short that Reincarnation exists, and memories are preserved, but only in very exceptional cases and perhaps after considerable care to awaken those memories is taken. While that view raises a lot of enthusiasm in many people, it must be noted that it is both self-limiting in its practical significance and still very difficult to evidence either for or against. To the best of my understanding it is such a view that Ian Stevenson hopes to have provided evidence for, although the statistical validity of his claims is strongly disputed.
Then there are the views typical of the French-Brazilian Spiritism that so plagues this accursed land where I live.
Hippolyte Rivail, a frenchman who lived in the troubled 19th century, convinced himself that he was a Druid reincarnate, styled himself as "Allan Kardec" accordingly, and wrote a half dozen books of unashamed wishful thinking that could make HPB pale by comparison. His "doctrine", for lack of a better word, is little more than a manic insistence that death is an illusion and that all people reincarnate time and again, in ever more refined and capable bodies, until they eventually attain enough "spiritual evolution" to "be together with God".
As beliefs go, it is really pissy-poor even when it is not actively toxic. It appeals to people who feel miserable and helpless and lack the means to learn better, or who are simply too emotionally immature to deal with reality in a healthy way. It is just a small, often-taken step away from decreeing that people should surrender to what I call "Green Lantern Syndrome", which is the compulsion to decide that reality is not to be taken seriously if one feels passionate enough about how they want things to be. Systematic self-delusion disguised as a religious faith, as a matter of fact.
It is little wonder that even the Spiritists themselves claim that most mental hospices hold a fair number of "mediums", as they call the people who can supposedly perceive "the other side".
Brazil is the last surviving stronghold of Kardecist Spiritism, with at least three million Spiritists in Brazil (almost 2% of our population), although their influence goes way beyond what that number would suggest. Their doctrine appeals to the Brazilian affinity with self-pity and delusions of grandeur, for Spiritists are all but assured that their "true merits" go far beyond that can be gauged in this world. It is a recurrent joke that everyone in Brazil is the reincarnation of some grand figure of some sort. Many people believe such claims, nonetheless. Which leads one to wonder what meaning there is to the whole idea of reincarnation. The obvious answer is: not much at all.