Update Nov 11, 2017. I've come across this old thread today and decided to expand my answer by showing the algorithm of the iterative fitting about which I was speaking initially.
Here is an iterative solution how to train a randomly simulated or preexistent variable $Y$ to correlate or covariate precisely as we desire (or very close to so - depending number of iterations) with a set of given variables $X$s (these cannot be modified).
Disclamer: This iterative solution I've found inferior to the excellent one based on finding the dual basis and proposed by @whuber in this thread today. @whuber's solution is not iterative and, more importantly for me, it seems to be affecting the values of the input "pig" variable somewhat less than "my" algorithm (it'd be an asset then if the task is to "correct" the existing variable and not to generate random variate from scratch). Still, I'm publishing mine for curiosity and because it works (see also Footnote).
$^1$ The correction formula can be further sophisticated, for example, to insure greater homoscedasticity (in terms of sums-of-squares) of $Y$ with every $X$ as well, simultaneously with attaining the correlations, - I've implemented a code for that too. (I don't know if such "double" task is solvable via a more neat - noniterative - approach such as @whuber's.)
