Actually, that's so easy it's unbelievable. I just did it :)
I don't know if it can be done with hooks or plugins or whatever (I don't know how to make them) but it can be done with a template edit.
1) Go to AdminCP > User Profile Manager > Add New User Profile Field > Single Line Text Box and Set the options as follows:
Title: Real Name
Description: Enter your Real Name here
(might as well leave all the other options default)
2) Click on Save and when you get taken to the User Profile Manager, make a note of what name it's been given (field5, field6, field7 etc.)
3) Go to Languages & Phrases > Phrase Manager > Add New Phrase > Set this up as follows:
Phrase Type: Global
Product: vBulletin
Varname: real_name
Test: Real Name
Click on Save
4) Go to Styles & Templates > Style Manager > Select Edit Templates from the All Style Options drop-down box > Find the postbit template > Look for:
<if condition="$post['field2']">
<div>$vbphrase[location_perm]: $post[field2]</div>
</if>
Below this add the following:
<if condition="$post['fieldX']">
<div>$vbphrase[real_name]: $post[fieldX]</div>
</if>
Where X is whatever number in the Profile Field Manager (e.g. field5 or field6 etc.
Then all you need to do is save!
Job's a good-un.
EDIT: You might also want to do the same for postbit_legacy template as well.