With MMPR (as for any PR system) when there are more than two popular parties there is of course an enhanced likelihood for minority, and coalition governments.
Depending on circumstances or, perhaps, one’s point of view, this can be either a plus or a minus: we get a variety of viewpoints represented and a need to work collaboratively with other parties, which, all in all, can be beneficial. But it can also lead to fragile or moribund governments that can’t get anything done.
There can be difficulties and delays in determining a coalition government as well, and constraints might be needed to trigger another election if a log-jam occurs such that a government is not established within a reasonable time.
The use of party-lists, whether closed (the party supplies the list in party-preferred order), open (the party supplies the list, but voters can affect the order based on voting within the party list), or other flexibility options, have their own respective virtues and flaws, as well as bringing different degrees of complexity to the vote.
This is fundamentally no different, however, than, as now, parties selecting individuals to run under their banner in constituency elections.
The simplest party-list solution from the voter’s standpoint is a party-supplied closed list (presumably determined by an electoral process within the party), though another automatic-list option might be worth looking at, such as automatically building the list from the party’s electoral-district candidates who were not elected.
As for accountability, while people appointed from party-lists are arguably still fundamentally beholden to their respective parties, this concern is mitigated given that the constituency members would be directly accountable to the voters in their respective constituencies.
“… By seeking to combine the best of the two major electoral principles, mixed member PR has clear attractions. But some fear it combines the worst of these two models, creating two classes of representatives with competing interests. Others are concerned about its relative complexity, and those suspicious of parties may have difficulty accepting the party’s role in establishing its list of candidates. Given its recent adoption in several nations, their experience with the system warrants close scrutiny.” — FairVote Canada, Factsheet #6
There remains another gaping flaw in this system, however, which is the single-member first-past-the-post (FPTP ) constituency elections themselves.
This means that, where there are more than two popular candidates in any district, as already extensively argued, the likelihood, as for any FPTP election, is increased that a candidate who is not preferred by the majority is elected. A later assignment of some number of seats based on a party-vote (which does not necessarily entail any voter input regarding the individuals appointed from the given party-list) does not correct this fundamental problem.
If, however, the district elections were done instead with Ranked Pairs, or by some other Condorcet method, it would eliminate this concern.
Also, depending on the specifics of the given approach, there can arise opportunities for gaming the system, such as: where a given party expects to win a large number of district seats such that they don’t expect to win many party-vote seats — they can encourage voters who vote for their constituency-candidates to cast their party-preference vote for a friendly other-party, possibly increasing the number of list-allocated members of such friendly other-party, and thus potentially increasing their allies in the legislature; this, while it can enhance legislative collegiality, and their ability to form coalitions, rather defeats the original purpose of the MMPR system.
Constraints that a party must satisfy in order to qualify for party-list allocation are sometimes implemented, intending to alleviate such problems — for example: earning a minimum percentage of the party-vote, or perhaps electing some minimum number of or proportion of district candidates, or both.