Double-counting coverage of overlapped read pairs
samtools depth is a simplified version of samtools mpileup, which handles overlapping regions by default. Since overlapping regions shouldn't be included in coverage calculations (after all, you're not covering a region twice) you can use samtools mpileup instead:

$ cat foo.sam
@SQ SN:1    LN:1000
read1   99  1   1   50  50M =   46  95  ATTTAAAAATTAATTTAATGCTTGGCTAAATCTTAATTACATATATAATT  &lt;&lt;&lt;&lt;&lt;&lt;&lt;&lt;&lt;&lt;&lt;&lt;&lt;&lt;&lt;&lt;&lt;&lt;&lt;&lt;&lt;&lt;&lt;&lt;&lt;&lt;&lt;&lt;&lt;&lt;&lt;&lt;&lt;&lt;&lt;&lt;&lt;&lt;&lt;&lt;&lt;&lt;&lt;&lt;&lt;&lt;&lt;&lt;&lt;&lt;  NM:i:0
read1   147 1   46  50  50M =   1   -95 ATTTAAAAATTAATTTAATGCTTGGCTAAATCTTAATTACATATATAATT  &lt;&lt;&lt;&lt;&lt;&lt;&lt;&lt;&lt;&lt;&lt;&lt;&lt;&lt;&lt;&lt;&lt;&lt;&lt;&lt;&lt;&lt;&lt;&lt;&lt;&lt;&lt;&lt;&lt;&lt;&lt;&lt;&lt;&lt;&lt;&lt;&lt;&lt;&lt;&lt;&lt;&lt;&lt;&lt;&lt;&lt;&lt;&lt;&lt;&lt;  NM:i:0

$ samtools mpileup -Q 1 foo.sam | cut -f 1,2,4
...
1   42  1
1   43  1
1   44  1
1   45  1
1   46  1
1   47  1
1   48  1
1   49  1
1   50  1
1   51  1
1   52  1
...


I haven't shown the entirety of the output and you'd want to use the -aa flag to ensure that all bases are returned.

Note that samtools mpileup is doing this internally by setting the base phred scores of overlapping bases in one of the mates to 0, which then get excluded due to -Q 1 (the default is -Q 13, which you'd want to change).

As an aside, you probably don't need exactly correct values, only approximates. Given that, you can use bamPEFragmentSize from deepTools (or a similar tool from another package) to get the median fragment and read length, determine the median number of covered bases from that and use it to get a quite accurate approximation of the actual coverage (multiply the median fragment base coverage by the number of fragments and divide by the genome size). That may prove faster and close enough.