William Goldman's article "What is a projective structure?" in the Notices of the AMS states that Daryl Cooper has shown that $\mathbb{RP}^3\#\mathbb{RP}^3$ has no $\mathbb{RP}^3$ structure. The article gives further references, but I couldn't find a reference for the Cooper result.
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Changed \\# to \# and RP to \mathbb{RP} (the second change was made so that enough characters were changed for the edit to be valid; normally I wouldn't make such an edit).
Michael Albanese
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Sam Nead
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