First of all, the problem might be skill based: the secret here is to go through the closing doors doing a barrel roll and trying to keep your ship in vertical position. For the PC version, there is no key for the “roll left” or right action, so go into Options-> Controls-> Set Keys & Buttons -> Airplane and you should find a “Roll Left” and “Roll Right” key. Bind one of them to something and hold it to keep the ship vertical. This is the secret to getting past those frustrating doors in the Real World mission.
However, this might not necessary be a skill-related issue. It appears that it’s a graphical issue, as strange as this might sound. The idea is that there’s a bug that results in doors being shut more quickly in low framerate situations. The fix? Simply lower your resolution to the lowest possible, play the mission and go through the doors (it will surely work!) and then switch the resolution back to what you were having.
I really hope that this helps get you past the closing doors in the Real World mission in Saints Row 4. I know it’s really frustrating – so make sure you also share this tip with your friends if they have the same problem.