Definitions: (click to show/hide)
Chameleon Chess: All pieces on the board which are displayed as orthodox Q, R, B, S, are Chameleons. A Pawn can promote only in Chameleonpieces. (Chameleon: On completing a move, a Chameleon (from classical standard type) changes into another piece, in the sequence QSBRQ… Promotion may be to a chameleon at any stage in the cycle).
Circe Chameleon: When a piece is captured it transforms according to the scheme S>B>R>Q>S, then it is replaced on its Circerebirth square (gamearray squares, of the same colour in the case of pieces, the file of capture in the case of pawns, the promotion square of the file of capture in the case of fairy pieces.).
Einstein Chess: All units (Ks excluded) change their type when they move, according to a precise pattern. For noncapture moves: Q>R, R>B, B>S, S>P, P remains P. For capturemoves: R>Q, B>R, S>B, P>S, Qremains Q.
Republican Chess (Type 2): There are no Kings : if the side which has played can put the opposite King on a square where it would be legally mate, then the opposite King is put on such a square. The opposite side can then put itself the other King on a square where it is mated.
No.1061 Michael Grushko Israel original – 04.05.2016

Solution: (click to show/hide)

white Pd3e4
black Pe6d5
hs#6 (2+2) Circe Chameleon Einstein Chess Chameleon Chess Republican Chess (Type 2)

1.e4*d5=S[+bPd7] e6e5 2.Sd5f6=B=S e5e4 3.d3*e4=S[+bPe7] e7*f6=S[+wBc1] {
} 4.Se4*f6=B=R[+bBf8] Bf8a3=R=B 5.Bc1b2=R=B Ba3*b2=R=Q[+wRa1] {
} 6.Rf6c6=Q=R[+bKb1] + Kb1*a1[+wQd1][+wKc1] # {
(C+ by Popeye 4.73)}

The combination of EinsteinChess and ChameleonChess seems absurd. Surely these conditions should be mutually exclusive? It seems that Popeye applies the Chameleon transformation first, then the Einstein transformation. For example, with 2.Sd5f6=B=S, the Knight becomes a Bishop by Chameleon rules, but as it is a noncapturing move it goes back to Knight by Einstein rules! If the conditions were applied in reverse order then the Knight would go to Pawn by Einstein rules, where it would remain.
The same thing happens with, for example, 5.Bc1b2=R=B. However in the capturing move 4.Se4xf6=B=R, the Knight goes to Bishop by Chameleon rules, then to Rook by Einstein rules.
“The combination of EinsteinChess and ChameleonChess seems absurd. Surely these conditions should be mutually exclusive?”
Not absurd. If the precedence is defined, from the purely logical point of view, the problems with this combination of conditions can be ok. They can very well be set up as correct puzzles.
On the other hand, possibility to be solved by human solver is close to zero, especially when it is spices by other fairy elements. And the quality from general chess composition viewpoints (strategy, logic, beauty of solution…), as is the case of many computer generated problem, is mostly zero too.
Regarding the question of precedence.
Perhaps precedence is not defined in the Popeye program.
It may be that the Popeye program has not a rigid/fixed/builtin order of precedence.
It may try giving each condition in turn precedence.
In the case of the present problem, perhaps using any order of precedence other than that used in the solution gives No Solution – as Popeye may have found out – after trying. Perhaps it would have shown other solutions had they existed.