Eilenberg-Zilber-type theorem for good fiber products? My question is:

If $p\colon X \to B$, $q\colon Y \to B$ are proper submersions, is there a characterization of $H_*(X \times_B Y)$ in terms of $H_*(X)$, $H_*(Y)$, $H_*(B)$ that is simpler than the Eilenberg-Moore spectral sequence?

Ideally, what I would like is a chain complex $A$ built from $C_*(X)$, $C_*(Y)$, $C_*(B)$ with a map $A \to C_*(X \times_B Y)$ which descends to an isomorphism.
But I'm willing to compromise -- it's OK if this map doesn't descend to an isomorphism.
Also, I don't care too much about what version of homology it is, and you can change the hypotheses on $p, q$ if you want (e.g. require them to be surjective).
 A: There are fibrations
\begin{align*}
 F \to X & \to B \\
 G \to Y & \to B \\
 F\times G \to X\times_B Y & \to B \\
 F\times G \to X\times Y & \to B\times B \\
 F \to X\times_BY &\to Y \\
 G \to X\times_BY &\to X
\end{align*}
and various comparison maps between them.  You are probably better off reasoning indirectly from the associated Serre spectral sequences rather than using anything like the Eilenberg-Moore spectral sequence.  Under mild assumptions the Serre spectral sequences will be finite-dimensional and have Poincaré duality on every page, and there will only be finitely many differentials.  The Eilenberg-Moore spectral sequence will typically have an infinitely generated $E_2$ page and infinitely many differentials, with no visible sign of Poincaré duality.  It is hard to see how you could avoid similar issues unless you can come up with a version of Tor groups that takes account of Poincaré duality, which seems hard.
A: I believe that the Eilenberg-Moore ss is just a calculational consequence of a result of the sort you are asking for.  Take a look at [Eilenberg, Samuel; Moore, John C.
Homology and fibrations. I. Coalgebras, cotensor product and its derived functors. Comment. Math. Helv. 40 1966 199–236]. 
A: Since you are interested in chain complexes related to those in the Eilenberg-Zilber Theorem you could look at Homological Perturbation Theory, once one of your surmersions is converted to a fibration.  
