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For a complex manifold equipped with an anti-holomorphic involution, which isreferred to as a real variety, the Smith-Thom inequality states that the total$\mathbb{F}_2$-Betti number of the real locus is not greater than the total$\mathbb{F}_2$-Betti number of the ambient complex manifold. A real variety iscalled maximal if the equality holds. In this paper, we present a series of newconstructions of maximal real varieties by exploring moduli spaces of certainobjects on a maximal real variety. Our results establish the maximality of thefollowing real varieties:  - Moduli spaces of stable vector bundles of coprime rank and degree over amaximal smooth projective real curve (known as Brugall\'e-Schaffhauser'stheorem, with a short new proof presented in this work); the same result holdsfor moduli spaces of stable parabolic vector bundles.  - Moduli spaces of stable Higgs bundles of coprime rank and degree over amaximal smooth projective real curve, providing maximal hyper-K\"ahlerexamples.  - If a real variety has non-empty real locus and maximal Hilbert square, thenthe variety itself and its Hilbert cube are maximal. This is always the casefor maximal real smooth cubic threefolds, but never the case for maximal realsmooth cubic fourfolds.  - Punctual Hilbert schemes on a maximal real projective surface withvanishing first $\mathbb{F}_2$-Betti number and connected real locus, such as$\mathbb{R}$-rational maximal real surfaces and some generalized Dolgachevsurfaces.  - Moduli spaces of stable sheaves on the real projective plane, or moregenerally, on an $\mathbb{R}$-rational maximal Poisson surface.  We also observe that maximality is a motivic property when interpreted asequivariant formality. Furthermore, any smooth projective real varietymotivated by maximal ones is also maximal.

Maximal real varieties from moduli constructions