Mature Husserl is usually seen as a mild anti-realist, but this is largely due to the maxim that phenomenology should be neutral (agnostic) on metaphysical matters because it is the data it produces that is to be used to adjudicate them later. This means that Husserl's observations are typically easy to adapt to a realist perspective, after all even a realist may find it methodologically meritorious to collect evidence without making realist presuppositions from the outset.
However, Husserl himself chose to embed phenomenology into the framework of transcendental idealism, and without the residual realism of Kant's "thing in itself". Here is from What is Wrong with Husserl's Scientific Anti-Realism? by Wiltsche:
"While some commentators criticize Husserl for his alleged scientific anti-ealism, others argue that Husserl’s position is much more realist than the first impression indicates. It is against this background that I want to argue for the following theses: a) The basic outlook of Husserl’sepistemology as well as his more substantial comments regarding the natural sciences indeed result in a sophisticated version of scientific anti-realism which bears certain esemblancesto Bas van Fraassen’s constructive empiricism..."
Be it as it may, Husserl focuses much more on how we come to experience space rather than on what it "is" (or is not). His idea f the genesis of our spatial experience is two-fold, first source is our sensory experiences associated with mobility, so-called kinaesthetic awareness, (construed not empirically but "internally", phenomenologically), this idea goes back to Kant's successor Herbart, who influenced Helmholtz, Riemann and Poincare on this issue; and second, coordination of moving activity with other subjects. For more details see Przywara's Husserl's and Carnap's Theories of Space:
"Husserl conceives the constitution of space as the monosubjective intentional transformation of manifold of so-called sensuous fields that we make during our corporal-moving activity (objective space is the correlate of the monosubjective transformation). On the other side he conceives it as intersubjective, intentional transformation of structures of subjective perceived spaces that we make (together with so-called ‘transcendental empathy’) within our community of transcendental subjects that communicate each other (hence, the objective space is the correlate of the intersubjective transformation)...
The ‘physical space’, in Husserl’s opinion, is only an intersubjective construct. Summarizing we can say that Husserlian conception is realistic (at least at the starting point) and moderate deterministic (to the moment of writing Ideas I and II, wherein Husserl starts reformulating his theory of space into the transcendental idealism)."
See also blog post The Construction of Thing and Space: An Introduction to Husserlian Phenomenology of Perception and Rizzo's chapter Thing and Space in Husserl, which comment on Husserl's 1907 lecture Thing and Space.