Our evidence is primarily genetic.

Human populations share between 1-4% of all their DNA with Neanderthals, which demonstrates conclusively that early homo sapiens interbred with Neanderthals, and thus led to the genetic makeup we see now, see here.

As far as archaeological remains are concerned, there is not a whole lot of early humans and Neanderthals just in general to work with, which is no surprise, really, because these kinds of archaeological remains rarely remain intact for the tens of thousands of years that it took to finally reach us in this modern world.

That being said, we have discovered some human remains which show evidence of this interbreeding as well, which demonstrates on genetic and archaeological grounds that homo sapiens and Neanderthals most certainly interbred and comingled, see here and here. These finds range from Portugal to Romania, and genetic testing has confirmed Neanderthal DNA in African populations as well.

In Italy a jawbone was discovered of a Neanderthal which shows striking similarities with homo sapiens jaws as well, indicating possible interbreeding there as well.

In short, yes, there is archaeological evidence for coexistence and interbreeding between anatomically modern humans and Neanderthals.