In the absence of Woodin cardinals, fine structural inner models for mild large cardinal hypotheses admit forcing extensions where bounded forcing axioms hold and yet the reals are projectively well-ordered.
@article{1120224728,
author = {Caicedo, Andr\'es Eduardo},
title = {Projective well-orderings and bounded forcing axioms},
journal = {J. Symbolic Logic},
volume = {70},
number = {1},
year = {2005},
pages = { 557-572},
language = {en},
url = {http://dml.mathdoc.fr/item/1120224728}
}
Caicedo, Andrés Eduardo. Projective well-orderings and bounded forcing axioms. J. Symbolic Logic, Tome 70 (2005) no. 1, pp. 557-572. http://gdmltest.u-ga.fr/item/1120224728/