The green cloth carrier is called a bandolier. You said ten round clips? The M1 Garand has 8 round clips called en bloc clips
, in which the rounds are in a staggered column. As far as I know, they were always one clip to bandolier pouch. The Springfield '03 used straight clips called stripper clips that held 5 rounds each
, and two of those would fit easily in a bandolier. Now there are 10 round stripper clips
for 7.62 NATO (.308 Winchester) for stripper clip loading of M14 magazines in situ. There were also 10 round stripper clips for the M1 Carbine
which have a sort of flat shovel end that acts as a guide..
If you have the 5 round stripper clips and .30-06 ammunition (individual cartridges about 3 1/3 inches long), and the box says M1, then I would assume it is M1 Ball ammunition which was obsoleted in 1944, when all .30-06 ammunition was corrosively primed. On the other hand, if you have pairs of 10 round clips for the M1 Carbine (individual cartridges about 1 5/8 inches long), that ammunition was always made with non-corrosive primers from the git go, and was the first U.S. military to have non-corrosive primers.