We give the rays (as vectors of length 20+1=21) of trop+Y(3,6) by modifying the positive tropical Grassmannian with a single vector, due to q = p123*p345*p156*p246-p234*p456*p126*p135.  That vector is in position 1 in the list of 15 vectors below.

The ordering in each vector is lexicographic: {1,2,3},{1,2,4},{1,2,5},{1,2,6},{1,3,4},{1,3,5},{1,3,6},{1,4,5},{1,4,6},{1,5,6},{2,3,4},{2,3,5},{2,3,6},{2,4,5},{2,4,6},{2,5,6},{3,4,5},{3,4,6},{3,5,6},{4,5,6}

In what follows we present s1,s2,...,s15.  The first vector s1 is new and corresponds to q; the others are positive tropical Plucker vectors (in the 20 indices 2,...,21).

{{1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0},{0,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0},{0,1,1,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0},{0,1,0,0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0},{0,-1,-1,-1,0,-1,-1,0,-1,0,0,-1,-1,0,-1,0,0,0,0,0,0},{0,-1,-1,-1,0,-1,-1,0,-1,0,0,0,0,0,0,0,0,0,0,0,0},{0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0},{0,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0},{0,-2,-2,-1,-1,-1,-1,-1,-1,-1,0,-1,-1,-1,-1,-1,0,0,0,0,0},{0,1,0,0,0,0,0,0,0,0,0,1,1,1,0,0,0,0,0,0,0},{0,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,1,0,0,0,0,0,0,0,0,0},{0,3,1,1,1,1,1,1,0,0,0,1,1,1,0,0,0,0,0,0,0},{0,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,-1,-1,-1,-1,-1,0,0,0,0,0},{0,-1,-1,-1,1,-1,-1,0,-1,0,0,-1,-1,0,-1,0,0,0,0,0,0},{0,0,-1,-1,0,-1,-1,0,-1,0,0,0,0,1,0,0,0,0,0,0,0}}

The pairs of compatible rays can be used to construct the fan.  They are as follows:

{{s1,s10},{s1,s11},{s1,s6},{s1,s9},{s1,s14},{s1,s8},{s1,s4},{s1,s12},{s1,s3},{s10,s11},{s10,s6},{s10,s4},{s10,s12},{s10,s3},{s10,s2},{s10,s15},{s10,s7},{s11,s6},{s11,s9},{s11,s3},{s11,s2},{s6,s9},{s6,s14},{s6,s4},{s6,s2},{s6,s15},{s6,s5},{s9,s14},{s9,s8},{s9,s3},{s9,s2},{s9,s5},{s9,s13},{s14,s8},{s14,s4},{s14,s5},{s8,s4},{s8,s12},{s8,s3},{s8,s7},{s8,s5},{s8,s13},{s4,s12},{s4,s15},{s4,s7},{s4,s5},{s12,s3},{s12,s7},{s3,s2},{s3,s7},{s3,s13},{s2,s15},{s2,s7},{s2,s5},{s2,s13},{s15,s7},{s15,s5},{s7,s5},{s7,s13},{s5,s13}}

Interpretation:

Fabcd = Eabc + Eabd + Eacd + Ebcd.
Gabcdef = Fabcd + Ecde + Ecdf.

s1 = Veronese,
s2 = E156,
s3 = F1234,
s4 = F1236,
s5 = E345,
s6 = F2345,
s7 = E123,
s8 = F3456,
s9 = F1256,
s10 = F1456,
s11 = E234 + E156,
s12 = E123 + E456,
S13 = G123456 (3-split),
s14 = E126 + E345,
s15 = G164523 (3-split).