There are several functorial constructions of the space BG for a topological group (meaning BG plus the universal G-bundle). First, there is the Milnor construction, treated in several textbooks. The Milnor construction is functorial and $EG \to BG$ is locally trivial for all topological groups. The Milnor construction is NOT monoidal in the sense that $B(G \times H) \cong BH \times BG$ ($B1$ is something like an infinite-dimensional simplex and not a point). On the other hand, there is the nerve-construction $BG:= |N_{\bullet} G|$ (plus a construction of $EG$). This is monoidal, but the map $EG \to BG$ is not always locally trivially (according to Graeme Segal, Classifying spaces and spectral sequences, p. 107). It is locally trivial if G is "locally well-behaved" (Segal gives a precise condition). Segal claims that if G is not locally well-behaved, then local triviality is not an appropriate concept. I would be happy to exclude groups like the p-adic integers from having a classiying space, but there are other groups which I do not like to throw away, like Homeo (X) for a manifold X (is this locally well-behaved??). Here is my question:

Is there a construction of $BG$, satisfying the following properties:

- functorial,
- monoidal,
- $EG \to BG$ is locally trivial,
- the class of groups to which it applies is "very large", including Homeo of reasonable spaces,
- Simple enough to be reasonably presented in a lecture course?