  We define a variant of intersection space theory that applies to many compact
complex and real analytic spaces $X$, including all complex projective
varieties; this is a significant extension to a theory which has so far only
been shown to apply to a particular subclass of spaces with smooth singular
sets. We verify existence of these so-called algebraic intersection spaces and
show that they are the (reduced) chain complexes of known topological
intersection spaces in the case that both exist. We next analyze "local duality
obstructions", which we can choose to vanish, and verify that algebraic
intersection spaces satisfy duality in the absence of these obstructions. We
conclude by defining an untwisted algebraic intersection space pairing, whose
signature is equal to the Novikov signature of the complement in $X$ of a
tubular neighborhood of the singular set.
