Four For Five

The Red Sox have scored an identical number of runs in each of their last five games:

                RS     RA
May 10 vs TBR    4      3
May 12 at LAA    4      3
May 13 at LAA    4      8
May 14 at LAA    4      5  (12 inn)
May 15 at SEA    4      5

That ties the team record, set in 1928:

                 RS    RA
August 1 at CWS  3      7  (2nd G)
August 2 at CWS  3      6
August 3 at CWS  3      1
August 4 at StL  3     11
August 5 at StL  3      1

[Thanks to SoSH's OttoC.]

***

I don't know what the all-time record is for most consecutive games scoring the same amount of runs, but here are lists of identical run totals from 1954-2009:

0 Runs: 4 games -- 7 times, last by 1992 Cubs

1 Run: 5 games -- 1962 Athletics, 1967 Twins, 1968 Dodgers, 2001 Tigers

2 Runs: 6 games -- 1980 Twins

3 Runs: 6 games -- 1993 Cleveland, 1984 Astros

4 Runs: 7 games -- 2003 Cleveland

5 Runs: 5 games -- 1964 Cubs, 1996 Cardinals, 2005 Dodgers

6 Runs: 5 games -- 1966 Phillies, 2001 White Sox

7 Runs: 6 games -- 1989 Brewers

8 Runs: 3 games -- 28 times, last by 2007 Royals; the 1997 Expos had a 1-2 record!

9 Runs: 3 games -- 18 times, last by 2008 Reds

10 Runs: 4 games -- 1977 Phillies

11 Runs: 3 games -- 1954 Giants, 1991 White Sox

12 Runs: 2 games -- 22 times, last by 2008 Twins

13 Runs: 3 games -- 1999 Cleveland

Naturally, the number of teams scoring at least a certain amount of runs will be higher. For example, two teams have scored at least eight runs in eight straight games.

And the 1993 Tigers scored at least 15 runs in three straight games, sweeping Baltimore in mid-August 15-1, 15-5 and 17-11. (Detroit then scored only one run in each of their next two games.)