I am a bit late to the party, but I just found this question. When I did research on the φ bond for What would follow in the series sigma, pi and delta bonds? I came across some predictions about possible φ bond in the ground state.
According to Gagliardi and Roos, there should be a φ bond in the ground state of the $\ce{U2}$ molecule. That being said, it is probably incredibly tough to synthesise this molecule.
Metal-metal multiple bonds have become fairly common as my earlier answer demonstrates.
It should be clarified that a quadruple bond does not necessarily mean that a δ bond is involved. Often enough the combination of the dz² metal orbitals lead to the formation of a second σ bond first.
It should further be explained, that a δ bond can be present, but the overall bond order could be lower than four.
Laura Gagliardi, Björn O. Roos, Nature, 2005, 433, 848-851. Available free at Archive ouverte UNIGE.
There is strong evidence to believe that no two elements in the periodic table can form a bond with greater order than 6
and links to onlinelibrary.wiley.com/doi/10.1002/anie.200603600/abstract. Guess it's relevant here. It may not make phi/gamma bonds impossible, but probably makes them rarer. $\endgroup$