As with the polyominoes, not very much is known about polyiamond reptiles but below are a few data mostly from Mike Reid.
rep-n² for all n
The diagram below shows rep-2² (and hence rep-4²) and rep-3² together with extensions from rep-k² to rep-(k+3)².
rep-n² for n = 6k, 6k-1, 6k+1
The diagrams below show rep-5² and rep-7² together with a extension from rep-n² to rep-(n+6)². (This is possible for all n since 6 x k parallelograms exist for all k>1).
The diagram below shows how to extend any rep-n² to a rep-(n+8)².
The data above for this octaimond are by no means complete.
One interesting reptile is the octiamond shown below which is rep-n² for all odd values of n but which is probably not rep-n² for any even n.
Self-replicating Pairs of Polyiamonds