Below you will find some references. I would also add a warning that these questions can become very difficult very soon.

Of course, one can base extend to the algebraic closure, and run the mmp there, but what guarantees that even the first step can be performed such that the resulting objects over the algebraic closure is the base extension of something reasonably related to the original object over the original field?

In particular, take a surface defined over $\mathbb R$ and let $X$ denote the corresponding surface over $\mathbb C$. Now if $X$ contains a $(-1)$-curve that has no real points, then when this curve is contracted, the real points, that is the original surface $X_{\mathbb R}$ does not change at all. That itself is of course not a problem and also one may argue that that $(-1)$-curve has to have a conjugate pair and contracting that results in a surface that should be again defined over $\mathbb R$ (In a different space but with the same set of real points as before). You can imagine that in higher dimensions even more complicated problems arise.

Another issue to keep in mind is that in real algebraic geometry people care about keeping the topology the same. Doing a flip screws that up, so the preferred way to do it is without flips. This seems like an impossible proposition, but Kollár's ingenious series of papers does exactly that. Well, actually what he does is that he performs flips on the complexified space but proved that it happens on the imaginary part and hence it does not mess with the topology of the real points. However, you still need to do it in order to keep the mmp going.

Anyway, to learn more about this, look at these papers:

MR1677128 (2000c:14078) Kollár, János Real algebraic threefolds. I. Terminal singularities. Dedicated to the memory of Fernando Serrano. Collect. Math. 49 (1998), no. 2-3, 335--360. (Reviewer: Mark Gross) 14P25 (14E30)

MR1639616 (2000c:14079) Kollár, János Real algebraic threefolds. II. Minimal model program. J. Amer. Math. Soc. 12 (1999), no. 1, 33--83. (Reviewer: Mark Gross) 14P25 (14E30)

MR1703903 (2000h:14049) Kollár, János Real algebraic threefolds. III. Conic bundles. Algebraic geometry, 9. J. Math. Sci. (New York) 94 (1999), no. 1, 996--1020. (Reviewer: Mark Gross) 14P25 (14E30)

MR1760882 (2001c:14087) Kollár, János Real algebraic threefolds. IV. Del Pezzo fibrations. Complex analysis and algebraic geometry, 317--346, de Gruyter, Berlin, 2000. (Reviewer: Mark Gross) 14P25 (14J30)

Another issue is the rational connectivity and its relation to Mori fiber spaces.

For this, again, one has to be very careful and state exactly what one wants. What notion of rational connectivity are you using? Are the curves defined over $\mathbb R$ or $\mathbb C$? Do you require $X_{\mathbb R}$ or $X$ (or both) be rationally connected?

To illustrate the difficulties, here is a conjecture of Nash (yes, *that* Nash): Let $Z$ be a smooth real algebraic variety. Then $Z$ can be realized as the real points of a *rational* complex algebraic variety.

This, actually, turns out to be false. Kollár calls it the shortest lived conjecture as it was stated in 1954 and disproved by Comessatti around 1914 (I don't remember the exact year). However, even if the statement is false, just the fact that it was made and no one realized for 50 years that it was false should show that these questions are by no means easy. (Comessatti's paper was in Italian and I have no idea how Kollár found it.)

Kollár showed more systematically the possible topology types of manifolds that can satisfy this statement. In particular, Kollár shows that any closed connected 3-manifold occurs as a possibly non-projective real variety birationally equivalent to ${\mathbb P}^3$. In other words the way Nash's conjecture fails is on the verge of the difference between projective and proper again showing that these questions are not easy.

Here are some references for Kollár's work on the Nash conjecture.

MR1641168 (99k:57038) Kollár, János The Nash conjecture for threefolds. Electron. Res. Announc. Amer. Math. Soc. 4 (1998), 63--73 (electronic). (Reviewer: A. Tognoli)

MR1865090 (2003a:14033) Kollár, János The topology of real and complex algebraic varieties. Taniguchi Conference on Mathematics Nara '98, 127--145, Adv. Stud. Pure Math., 31, Math. Soc. Japan, Tokyo, 2001. (Reviewer: Sándor J. Kovács)

MR1882536 (2002m:14046) Kollár, János The topology of real algebraic varieties. Current developments in mathematics, 2000, 197--231, Int. Press, Somerville, MA, 2001. (Reviewer: Grigory B. Mikhalkin) 14P25

MR1941579 (2004c:14117) Kollár, János The Nash conjecture for nonprojective threefolds. Symposium in Honor of C. H. Clemens (Salt Lake City, UT, 2000), 137--152, Contemp. Math., 312, Amer. Math. Soc., Providence, RI, 2002. (Reviewer: Sándor J. Kovács)