My definition for Realcompact Space: $X$ is realcompact if it can be embedded as a closed subspace of a product of copies of the real line.
I found the following characterisation of realcompact spaces in Gillamn and Jerison's 'Rings of Continuous Functions' $-$
X is realcompact iff every prime z-filter with countable intersection property is fixed.
There, the proof is given in terms of ideals - a concept I'm not acquainted with. Nor am I acquainted with compactifications. So, is it possible to give a proof of this just in terms of filters and ultrafilters?