Non-rooted MST of directed graph
If I understand your question correctly you aren't looking for a tree, or an arborescence, but rather a minimum weight set of arcs that suffice to keep the graph strongly connected.

Your problem is NP-complete by reduction from Hamiltonian cycle problem. Given an input undirected graph that you wish to find the Hamiltonian cycle of create a directed graph by replacing each edge with a pair of oppositely directed arcs. Give all arcs weight 1. This directed graph has a set of arcs that leave the graph strongly connected with weight at most the number of vertices if and only if it has a Hamiltonian cycle.

You can get a two-approximation by picking an arbitrary root, finding the minimum weight spanning tree with arcs pointing towards the root, finding the minimum weight spanning tree with arcs pointing away from the root, and then unioning the two sets of arcs.

I don't recognize this problem off the top of my head but have you tried looking in the various NP-complete problem directories for it?