Warren D. Smith, Nov. 2015. This system was invented by Jameson Quinn. WDS has however edited it and also changed it in ways JQ might not approve, the system here is not quite the same as JQ's.
Here is one concrete system suggestion for Canada. It incorporates ingredients of score and approval voting, asset voting, and open-party-list voting, all merged, plus some new thoughts, into one hopefully-lovely, beautifully engineered whole (although its detractors might brand it an "ugly mutt").
EM=(Extra Member=MMP; 2/3 of the seats are determined locally by votes from a given riding/district; the other 1/3 are "extra members" allocated regionally to ensure a proportional result.
FA=Free Adherence=opt in to other riding; By default, you vote in your own district. But if you don't like any of the candidates in your district, you are always free to cast your vote in some other district. (If you do, that automatically makes you vote "2" for NOTA in your home riding, see "CS" below.) The more voters choose to vote in a district, the more likely that district will get extra members.
TI=Transfer Independently=delegated; Your vote automatically helps make sure your favorite party gets its fair share of seats. It works as follows. After all the parties all have their minimum fair share, each party usually will still have some leftover votes. Parties with the smallest leftovers will pass them to those with bigger leftovers. Each voters' preferred candidate is the one who decides how that vote is transferred; in other words, for transfers, the vote is delegated.
CS=Combined Score=score open list) Voter can give each candidate in her district (or whichever district you choose to vote in) a score of 0, 1 , or 2, corresponding to unapproved, approved/acceptable, or preferred. At most one score of 2 is allowed. Within each district, the highest total score (counting only "local votes," i.e. cast by residents of that district only) wins. In each district there always is one additional pseudo-candidate on ballot called "none of the above" or NOTA. If NOTA wins his local seat, then no real candidate wins it and that riding's seat instead is allocated as an "extra member" seat. After the local winners are announced, then, within each party, the highest scorers (now based on all votes, both local and nonlocal) are first in line for the "Extra Member" seats.
To illustrate this, consider the US state of Arizona. Its population is enough for 9 seats in the House of Representatives. Currently, that means it has 9 districts.
(You can see the effects of "gerrymandering" in the bizarrely-shaped peninsulas and isthmuses of these districts, especially the 4th and 9th.)
Under EMFATICS, it would have 6 districts and 3 "extra seats". Each party would get a seat (local or extra) for every full 10% of ballots which preferred it (that is, 100% divided by the number of seats plus one). Say Republicans scored highest in 4 of the district races, and Democrats scored highest in 2. Meanwhile, the overall breakdown of preferred votes (scores of 2) was:
43% Republican (4 seats; already taken; remainder 3%)
36% Democratic (3 seats; 2 taken; remainder 6%)
11% Libertarian (1 seat; remainder 1%)
7% Independent (remainder 7%) (Note: this is a single candidate.
If there were several independents, each would be considered as their own separate party).
3% Green (remainder 5%)
After the local winners are seated, the Democrats and the Libertarians each have enough votes for one extra seat; these go to the highest-scoring candidates available in each case. Now the non-winning candidates from the party with the lowest remainder – the Libertarians – must choose how transfer their remainders to other parties. Each non-seated Libertarian candidate controls a number of votes proportional to how many voters preferred them; they decide which party to give to. For simplicity, let's say they all choose the Republicans, so the Republicans go from 3% remainder to 4%.
The next lowest remainder is now the Greens. Let's say one Green gives her 1% of votes to the Independent (bringing him to 8%), while the other Greens give their 2% to the Democrats (bringing them to 8% too).
The next lowest remainder is the Republicans. Let's say they all give their 4% to the Independent. That brings him to 12%, more than enough for the final seat. He is seated, and the process is complete.
So, the final totals are: 4R, 3D, 1L, 1I.
Fully proportional: Almost all voters are guaranteed that their vote will help elect a representative that they supported, directly or indirectly. In the example above, only the 3% Greens went entirely unrepresented.
Voters decide: There are no "safe seats" that are not accountable to the voters. Because a party's extra seats always go to whichever of its candidates have the highest scores, and because voters always have the option to vote in another district if there are no good candidates locally, corrupt party insiders have no way to guarantee reelection or shield themselves from accountability.
Simple ballots: You never need to (although you can) look at more than one district worth of candidates. Any voter who votes one candidate as "preferred" can be assured that their vote will help get that candidate's party fair representation.
Independents get a fair chance: Even if an independent doesn't quite win their district, they still have a fair chance to get one of the "extra member" seats. But that choice is made by actual candidates from other parties choosing to give their remainder votes to the independent, not just by a mathematical formula. So a centrist independent with crossover appeal who got 45% preferences locally might be enough to get a seat; but an extremist with the same local total would not.
More about independents: suppose there were no parties. Everybody ran independent. In that case things still should work out pretty decently, it'd basically be "asset voting" used as a top-up for "approval voting."
Looking up numeric codes for out-of-district voting (or acquiring ballots in other ridings): This may be too difficult for voters, and the fact many races are generally printed on the same ballot for a riding (not just the federal MP race) makes it difficult to use the "just use the foreign riding's ballot" solution. Also many people vote by mail (including in Canada), complicating matters further.
Independents would presumably be unfairly disadvantaged in this system, as it might be unrealistic to list all of them as well as the parties, and the explicit use of parties in this system causes a built-in bias against independents, as well as offending some people ("as a matter of principle, voting systems should not ever need to refer to parties"). In its defense, though, EMFATICS does try keep its party-usage low and still ought to produce decent results even if there were no parties and all candidates ran independent.
With only "two and a half" score levels (0,1,2 with at most one use of 2 per voter) EMFATICS inherently loses accuracy and expressiveness versus systems employing score voting with, say, 0,1,2,3,4,5,6,7,8,9 as allowed scores. It is known that the voters in some real elections genuinely needed at least 7 score levels to express their opinions, and that approval voting has substantially worse Bayesian regret than score voting.
The plurality-like nature of the single "2" score means that EMFATICS suffers all the same evils as plurality voting had, albeit in a form lessened in severity by about a factor of 2. Namely, we can have "vote splitting" where, say, 3 Reds run versus a Blue, and the Reds "split the vote" and hence lose to the Blue, even though the Red view was more popular. Consequently EMFATICS is vulnerable to candidate-cloning. For essentially the same reason, EMFATICS is not "favorite-safe," i.e. it can be a strategic mistake for a voter to honestly give her "2" to her favorite. There also is conflict between the desire to give your 2 to the best local, versus the desire to give your 2 to the best party, which is a different incentive for voters to lie and varies in different ridings (e.g. in a riding where Joe was expected to win safely, the incentive is to use your 2 for a different purpose).
It is possible that NES strategy by voters about the 2's might yield enough artificial advantage for the top 2 parties to ultimately engender 2-party domination (again in a manner resembling how plurality-voting eventually engendered massive and apparently permanent 2-party domination in the USA).
All the criticisms in the preceding 3 paragraphs could be lessened in severity if EMFATICS were redesigned as follows: the allowed scores are now {0,1,2,...,M-1,M} where at most a single "M" (max) score is allowed per voter (M plays the role that "preferred" played in the original design). Then make M fairly large, such as M=10.