Supplementary Figure S1: Separated benthic 18 O data over 5 Myr. (a) Benthic LR04 benthic 18 O stack 16 ( ) in black with seawater 18 O ( w ) in blue

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1 Supplementary Figure S1: Separated benthic 18 O data over 5 Myr. (a) Benthic LR04 benthic 18 O stack 16 ( ) in black with seawater 18 O ( w ) in blue and temperature contribution ( T ) in green. (b) Surface-air temperature anomaly relative to PD ( C). (c) Sea level derived from ice volume (m) in black. Ice volume (m s.e.) from the NaIS in blue, the EuIS in red, the AIS in orange and the GrIS in green. Thick lines in panel a-c represent the 400-kyr running mean. 1

2 Supplementary Figure S2: Wavelets over 5 Myr. Wavelet analysis 48 of: (a) January 80 S Insolation 43. (b) The January 80 S anomaly relative to PD calculated as max(0, Q80S). (c) Simulated sea level derived from global ice volume. (d) Sub-shelf oceanic melt of the AIS. (e) Total ice volume of the AIS. (f) Ice discharge towards the open ocean from the AIS. 2

3 Supplementary Figure S3: Power evolution. For all panels, 400-kyr power is shown in black, 100-kyr power in red and 41-kyr power in blue. (a) LR04 benthic 18 O stack 16. (b) Surface-air temperature anomaly. (c) Leg 138 benthic 13 C (ref. 13). (d) NH ice volume. (e) AIS ice volume, with 400-kyr evolution without insolation variability in the weighting function equation (7) in grey, on the same scale as panel (d). (f) Mediterranean stack planktonic 13 C (ref. 13). Power evolution performed with AnalySeries 49 using a moving window of 1 Myr steps of 10 kyr and a Parzen window with 90% lags. 3

4 Supplementary Figure S4: Reconstruction over 5 Myr. (a) Total NH ice volume (10 6 km 2 ). (b) Antarctic ice volume (10 6 km 2 ) in orange and in black the volume without insolation variability in equation (7). (c) Antarctic ice discharge (10 12 kg yr -1 ) in orange and in black the discharge without insolation variability in equation (7). (d) The January 80 S anomaly relative to PD 43. 4

5 Supplementary Figure S5: Uncertainty in model results. In all panels the grey band represent the minimum and maximum value of the sensitivity runs given in Supplementary Table S2. The colour band gives a 2 value around the default run 6, which is shown by the black line. (a) Temperature. (b) Sea level, (c) AIS volume and (d) GrIS volume. 5

6 Supplementary Figure S6: Comparison with Sea Surface Temperatures. In all panels the blue curve is the modelled surface-air temperature anomaly, the red line represents the reconstructed Sea Surface Temperature (SST) from: (a) North Atlantic Site 982 SST 50. (b) Equatorial Pacific Site 846 SST 33 and (c) Southern Ocean Site 1090 SST 51. Thick lines represent the 400-kyr running mean. 6

7 Supplementary Figure S7: Comparison of deep-water temperature. (a) A comparison of the original benthic 18 O data, in blue the LR04 benthic stack 16 and in red the data from site 1123, Chatham Rise, east of New Zealand 29. (b) Reconstructed deep-water temperature anomaly from ANICE in blue and proxy derived data from site 1123, using the benthic 18 O data and Mg/Ca deep-water temperature proxies 29. 7

8 Supplementary Figure S8: Wavelets of runs without insolation in sub-shelf melt. Wavelet analysis 48 of: (a) Sub-shelf oceanic melt of the AIS. (b) Total ice volume of the AIS. (c) Ice discharge towards open ocean from the AIS. For comparison with Supplementary Fig S2d-f. 8

9 Supplementary Table S1: Sensitivity tests. The default run is marked in bold. Model run E SIA E SSA Rest-C abl GrIS-C abl

10 Supplementary Table S2: Uncertainties from runs in Supplementary Table S1, the standard deviation ( ), the mean maximum value, the mean minimum value, and the maximum and minimum deviation over the entire run. All values are relative to the default run 6. Variable Mean max. Mean min. Max. dev. Min. dev. T surf ( C) Sea level (m) EuIS (m s.e.) NaIS (m s.e.) AIS (m s.e.) GrIS (m s.e.)

11 Supplementary Table S3: Model parameters. Constant & description Value i Ice density (kg m -3 ) 910 fw Fresh water density (kg m -3 ) 1000 w Seawater density (kg m -3 ) 1028 c po Specific heat capacity of ocean (J kg -1 C -1 ) 3974 T Thermal exchange velocity (m s -1 ) 10-4 F melt Sub ice-shelf melt parameter (m s -1 ) L Latent heat of fusion (J kg -1 ) T oc CD, PD and WM sub ice-shelf temperature ( C) -5, -1.7, 2 11

12 Supplementary References 50. Lawrence, K. T., Herbert, T. D., Brown, C. M., Raymo, M. E. & Haywood, A. M. High amplitude variations in North Atlantic sea surface temperature during the early Pliocene warm period. Paleoceanography 24, PA2218 (2009). 51. Martínez-Garcia, A., Rosell-Melé, A., McClymont, E. L., Gersonde, R. & Haug, G. H. Subpolar link to the emergence of the modern equatorial Pacific cold tongue. Science 328, (2010). 12