JavaScript (ES6), 60 bytes
f=(b,...a)=>eval(b.join`+`)?f(...a,b.slice(1),b.pop()?b:b):b
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Commented
f = ( // f is a recursive function taking:
b, // b[] = next array
...a // a[] = list of all other arrays
) => //
eval(b.join`+`) ? // if the sum of b[] is not zero:
f( // do a recursive call:
...a, // first try with the other arrays that were
// already passed to this iteration
b.slice(1), // then try with b[] without the leading term
b.pop() ? b : b // and finally with b[] without the trailing term
// NB: we don't mind modifying b[] at this point,
// and this is shorter than b.slice(0, -1)
) // end of recursive call
: // else:
b // success: return b[]
